Problem Solving
Solve problems with or without a calculator
Interpret a calculator display of 4.5 as £4.50 in
context of money
Use a calculator and inverse operations to find
missing numbers, including decimals as for
example:
6.5 – 9.8 = □
4.8 ÷ □ = 0.96
1/8 of □ = 40
Use inverses to check results, for example,
703/19 = 37 appears to about right because 36 x
20 = 720
Carry out simple calculations involving negative
numbers in context
Level 4
What would 0.6 mean on a calculator display if
the units were £s, metres, hours, cars?
What is the important information in this
problem?
Show me a problem that you would use a
calculator to work out the answer. Show me a
problem that you wouldn’t use a calculator? How
do you decide?
Is it always quicker to use a calculator?
What key words tell you that you need to add,
subtract, multiply or divide?
How would you use a calculator to solve this
problem?
Choose a number to put into a calculator. Add
472 (or multiply by 26) what single operation will
get you back to your starting number?
Will this be the same for different starting
numbers? How do you know?
Understand and use an appropriate non-calculator method for
Level 5
solving problems that involve multiplying and dividing any
three digit number by any two-digit number
Show how you could work these out without a
Give pupils some examples of multiplications and
calculator:
divisions with mistakes in them. Ask them to
 348 × 27
identify the mistakes and talk through what is
 309 × 44
wrong and how they should be corrected.
 19 × 423
Explain your choice of method for each
Ask pupils to carry out multiplications using the
calculation.
grid method and a compact standard method.
Ask them to describe the advantages and
Find the answer to each of the following, using a disadvantages of each method.
non-calculator method.
How do you go about estimating the answer to a
 207  23
division?
 976  61
 872  55
317 people are going on a school coach trip.
Each coach will hold 28 passengers. How many
coaches are needed?
611 is the product of two prime numbers. One of
the numbers is 13. What is the other one?
Solve simple problems involving ordering, adding, subtracting negative numbers in context
Immediately before Sharon was paid, her bank
‘Addition makes numbers bigger.’ When is this
balance was shown as -£104.38; the minus sign statement true and when is it false?
showed that her account was overdrawn.
Immediately after she was paid, her balance was Subtraction makes numbers smaller.’ When is
£1312.86. How much was she paid?
this statement true and when is it false?
The temperatures in three towns on January 1st
were:
Apton
-5°C
Barntown
2°C
Camtown
-1°C
 Which town was the coldest?
 Which town was the warmest?
 What was the difference in temperature
between the warmest and coldest towns?
The answer is -7. Can you make up some
addition/subtraction calculations with the same
answer.
The lowest winter temperature in a city in
Canada was -15°C. The highest summer
temperature was 42°C higher. What was the
difference in temperature between the minimum
and the maximum temperature?
Talk me through how you found the answer to
this question.
The answer on your calculator is -144. What
keys could you have pressed to get this answer?
How does a number line help when adding and
subtracting positive and negative numbers?
Apply inverse operations and approximate to check answers
Level 5
to problems are of the correct magnitude
Discuss questions such as:
Looking at a range of problems or calculations,
ask:
 Will the answer to 75 ÷ 0.9 be smaller or
 Roughly what answer do you expect to
larger than 75?
get?
 How did you come to that estimate?
Check by doing the inverse operation, for
 Do you think your estimate is higher or
example:
lower than the real answer?
Use a calculator to check :

Explain your answers.
43.2 x 26.5 = 1144.8 with 1144.8  43.2
3
of 320 = 192 with 192 x 5  3
5
How could you use inverse operations to check
that a calculation is correct? Show me some
3  7 = 0.4285714…with 7 x 0.4285714
examples.
Calculate percentages and find the outcome of a given
Level 6
percentage increase or decrease
Use written methods, e.g.
Talk me through how you would
 Using an equivalent fraction: 13% of 48;
increase/decrease £12 by, for example 15%. Can
13/100 × 48 = 624/100 = 6.24
you do it in a different way? How would you find
 Using an equivalent decimal: 13% of 48;
the multiplier for different percentage
0.13 × 48 = 6.24
increases/decreases?
 Using a unitary method: 13% of 48; 1% of 48
= 0.48 so 13% of 48 = 0.48 × 13 = 6.24
The answer to a percentage increase question is
£10. Make up an easy question. Make up a
Find the outcome of a given percentage
difficult question.
increase or decrease. e.g.
 an increase of 15% on an original cost of £12
gives a new price of £12 × 1.15 = £13.80,
 or 15% of £12 = £1.80 £12 + £1.80 =
£13.80
Make and justify estimates and approximations of calculations;
Level 7
estimate calculations by rounding numbers to one significant
figure and multiplying and dividing mentally
Examples of what pupils should know and be
Probing questions
able to do
Estimate answers to:
Talk me through the steps you would take to find
5.16 x
an estimate for the answer to this calculation?
0.0721 x
(186.3 x 88.6)/(27.2 x 22.8)
Would you expect your estimated answer to be
greater or less than the exact answer? How can
you tell? Can you make up an example for
which it would be difficult to decide?
Show me examples of multiplication and division
calculations using decimals that approximate to
60.
Why is 6 ÷ 2 a better approximation for 6.59 ÷
2.47 than 7 ÷ 2?
Why it is useful to be able to estimate the answer
to complex calculations?
Use fractions or percentages to solve problems involving
Level 8
repeated proportional changes or the calculation of the
original quantity given the result of a proportional change
Solve problems involving, for example
Talk me through why this calculation will give the
compound interest and population growth using
solution to this repeated proportional change
multiplicative methods.
problem.
Use a spreadsheet to solve problems such as:
How long would it take to double your
investment with an interest rate of 4% per
annum?
A ball bounces to ¾ of its previous height each
bounce. It is dropped from 8m. How many
bounces will there be before it bounces to
approximately 1m above the ground?
How would the calculation be different if the
proportional change was…?
Solve problems in other contexts, for example:
Each side of a square is increased by 10%. By
what percentage is the area increased?
The length of a rectangle is increased by 15%.
The width is decreased by 5%. By what
percentage is the area changed?
Give pupils a set of problems involving repeated
proportional changes and a set of calculations.
Ask pupils to match the problems to the
calculations.
What do you look for in a problem to decide the
product that will give the correct answer?
How is compound interest different from simple
interest?
Solve problems involving calculating with powers, roots and numbers expressed in standard form,
checking for correct order of magnitude and using a calculator as appropriate
Use laws of indices in multiplication and division, Convince me that
for example to calculate:
37 x 32 = 39
37 ÷ 3-2 = 39
37 x 3-2 = 35
What is the value of c in the following question,
48 X 56 = 3 x 7 x 2c
Understand index notation with fractional
powers, for example knowing that n1/2 = √n and
n1/3 = 3√n for any positive number n.
Convert numbers between ordinary and
standard form. For example:
734.6 = 7.346 x 102
0.0063 = 6.3 x 10-3
Use standard form expressed in conventional
notation and on a calculator display. Know how
to enter numbers on a calculator in standard
form.
Use standard form to make sensible estimates
for calculations involving multiplication and
division.
Solve problems involving standard form, such
as:
Given the following dimensions
Diameter of the eye of a fly:
8 × 10-4 m
Height of a tall skyscraper:
4 × 102 m
Height of a mountain
8 × 103 m
How many times taller is the mountain than the
skyscraper?
How high is the skyscraper in km?
When working on multiplications and divisions
involving indices, ask:
Which of these are easy to do? Which are
difficult? What makes them difficult?
How would you go about making up a different
question that has the same answer?
What does the index of ½ represent?
What are the key conventions when using
standard form?
How do you go about expressing a very small
number in standard form?
Are the following always, sometimes or never
true:
 Cubing a number makes it bigger
 The square of a number is always positive
 You can find the square root of any number
 You can find the cube root of any number
If sometimes true, precisely when is it true and
when is it false?
Which of the following statements are true?

163/2 = 82

Length of an A4 piece of paper is 2.97 x
10-5km



8-3 =
272 = 36
= 5√7
Co-ordinates and Graphs
Use and interpret coordinates in the first quadrant
Given the coordinates of three vertices of a
rectangle drawn in the first quadrant, find the
fourth
Level 4
What are the important conventions when
describing a point using a coordinate?
I’m thinking of a co-ordinate that I want you to
plot. I can only answer ‘yes’ and ‘no’. Ask me
some questions so you can plot the coordinate.
How do you use the scale on the axes to help
you to read a co-ordinate that has been plotted?
How do you use the scale on the axes to help
you to plot a co-ordinate accurately?
If these three points are the three vertices of a
rectangle how will you find the coordinates of the
fourth point?
Use and interpret coordinates in all four quadrants
Level 5
Plot the graphs of simple linear functions.
If I wanted to plot the graph y = 2x how should I
Generate and plot pairs of co-ordinates for
start?
y = x + 1, y = 2x
How do you know the point (3, 6) is not on the
Plot graphs such as: y = x, y = 2x
line y = x + 2?
Plot and interpret graphs such as y = x,
y = 2x , y = x + 1, y = x - 1
Can you give me the equations of some graphs
that pass through (0, 1)? What about…?
Given the coordinates of three points on a
straight line parallel to the y axis, find the
equation of the line.
How would you go about finding coordinates for
this straight line graph that are in this quadrant?
Given the coordinates of three points on a
straight line such as y = 2x, find three more
points in a given quadrant.
Plot the graphs of linear functions, where y is given explicitly in terms of x;
Level 6
recognise that equations of the form y = mx+c correspond to straight-line graphs
Plot the graphs of simple linear functions using
How do you go about finding a set of
all four quadrants by generating co-ordinate
coordinates for a straight line graph, for example
pairs or a table of values. e.g.
y = 2x + 4?
 y = 2x - 3
 y = 5 - 4x
How do you decide on the range of numbers to
put on the x and y axes?
Understand the gradient and intercept in y = mx
+ c, describe similarities and differences of given How do you decide on the scale you are going to
straight line graphs. e.g.
use?
 y = 2x + 4
 y = 2x - 3
If you increase/decrease the value of m, what
effect does this have on the graph? What about
changes to c?
Without drawing the graphs, compare and
contrast features of graphs such as:
 y = 3x
 y = 3x + 4


y=x+4
y=x–2


y = 3x – 2
y = -3x + 4
What have you noticed about the graphs of
functions of the form y = mx + c? What are the
similarities and differences?
Construct functions arising from real-life problems and plot their corresponding graphs;
interpret graphs arising from real situations
The graph below shows information about a race What do the axes represent?
between two animals – the hare (red) and the
tortoise (blue)
In the context of this problem does every point
on the line have a meaning? Why?
What does this part of the graph represent?
What does this point on the graph represent?
What sort of questions could you use your graph
to answer?



Who was ahead after 2 minutes?
What happened at 3 minutes?
At what time did the tortoise draw level with
the hare?
 Between what times was the tortoise
travelling fastest?
By how much distance did the tortoise win the
race?
For real-life problems that lead to linear
functions:
 How does the gradient relate to the original
problem?
 Do the intermediate points have any practical
meaning?
 What’s the relevance of the intercept in
relation to the original problem?
Plot graphs of simple quadratic and cubic functions
Construct tables of values, including negative
values of x, and plot the graphs of these
functions.
 y = x²
 y = 3x² + 4
 y = 2x2 – x + 1
 y = x³
Level 7
How can you identify a quadratic function from
its equation? What about a cubic function?
How do you find an appropriate set of
coordinates for a given quadratic function?
Convince me that there are no coordinates on
the graph of y=3x²+4 which lie below the x-axis.
Why does a quadratic graph have line
symmetry? Why doesn’t a cubic function have
line symmetry? How would you describe the
symmetry of a cubic function?
Understand the effect on a graph of addition of (or multiplication by) a constant
Given the graph of y=x2, use it to help sketch the
graphs of y=3x2 and y=x2+3
Explore what happens to the graphs of the
functions, for example:
y = ax2 + b for different values of a and b
y = ax3 + b for different values of a and b
y = (x ± a)(x ± b) for different values of a and b
Level 8
Show me an example of an equation of a graph
which moves (translates) the graph of y=x³
vertically upwards (in the positive y-direction)
What is the same/different about: y=x², y=3x²,
y=3x²+4 and 1/3x²
Is the following statement always, sometimes,
never true: As 'a' increases the graph of y=ax²
becomes steeper
Convince me that the graph of y=2x² is a
reflection of the graph of y=-2x² in the x-axis
Sketch, identify and interpret graphs of linear, quadratic,
cubic and reciprocal functions, and graphs that model real situations
Show me an example of an equation of a
Match the shape of graphs to given relationships,
quadratic curve which does not touch the xfor example:
axis. How do you know?
 the circumference of a circle plotted against its
Show me an example of an equation of a
diameter
parabola (quadratic curve) which
 the area of a square plotted against its side
 has line symmetry about the y-axis
length
 does not have line symmetry about the y the height of fluid over time being poured into
axis
various shaped flasks
How do you know?
Interpret a range of graphs matching shapes to
What can you tell about a graph from looking at
situations such as:
The distance travelled by a car moving at constant its function?
speed, plotted against time;
Show me an example of a function that has a
The number of litres left in a fuel tank of a car
graph that is not continuous, i.e. cannot be
moving at constant speed, plotted against time;
drawn without taking your pencil off the paper.
The number of dollars you can purchase for a
Why is it not continuous?
given amount in pounds sterling;
The temperature of a cup of tea left to cool to
How would you go about finding the y value for
room temperature, plotted against time.
a given x value? An x value for a given y
value?
Identify how y will vary with x if a balance is
arranged so that 3kg is placed at 4 units from the
pivot on the left hand side and balanced on the
right hand side by y kg placed x units from the
pivot
Constructing and Using Formula
Begin to use formulae expressed in words
Explain the meaning of and substitute integers
into formulae expressed in words, or partly in
words, such as the following:

number of days = 7 times the number of
weeks
 cost = price of one item x number of
items
 age in years = age in months ÷ 12
 pence = number of pounds × 100
 area of rectangle = length times width
 cost of petrol for a journey
= cost per litre × number of litres used
Level 4
Show me an example of a formula expressed in
words
How can you change ‘Cost of Plumber’s bill =
£40 per hour’ to include a £20 call-out fee.
I think of a number and add twelve – do you
know what my number is? Why or why not?
'I think of a number and add 12. The answer is
17.' Do you know what my number is? Why?
Use formulae expressed in words, for example
for a phone bill based on a standing charge and
an amount per unit
Recognise that a formula expressed in words
requires an equals symbol, for example,


'I think of a number and double it',
is different from
'I think of a number and double it. The
answer is 12'.
Give pupils two sets of cards, one with formulae
in words and the other with a range of
calculations that match the different formulae
(more than one for each formula in words). Ask
them to sort the cards.
How do you know this calculation is for this
rule/formula)?
Why is it possible for more than one calculation
to match with the same rule? Could there be
others?
What’s the same and what’s different about the
calculations for the same rule/formula?
Construct, express in symbolic form, and use simple formulae
Level 5
involving one or two operations
Use letter symbols to represent unknowns and
How do you know if a letter symbol represents
variables.
an unknown or a variable?
Understand that letter symbols used in algebra
stand for unknown numbers or variables and not What are the important steps when substituting
labels, e.g. ‘5a’ cannot mean ‘5 apples’
values into this expression/formula?
What would you do first? Why?
Know and use the order of operations and
How would you continue to find the answer?
understand that algebraic operations follow the
same conventions as arithmetic operations
How are these two expressions different?
Recognise that in the expression 2 + 5a the
multiplication is to be performed first
Understand the difference between expressions
such as:
 2n and n +2
 3(c + 5) and 3c + 5
 n² and 2n
 2n² and (2n)²
Simplify or transform linear expressions by
collecting like terms; multiply a single term over
a bracket.
Simplify these expressions:
 3a + 2b + 2a – b
 4x + 7 + 3x – 3 – x
 3(x + 5)
 12 – (n – 3)
 m(n – p)
 4(a + 2b) – 2(2a + b)
Substitute integers into simple formulae, e.g.
Find the value of these expressions when a = 4.
3a2 + 4
2a3
Find the value of y when x = 3
y = 2x + 3
x
y=x-1
x +1
Simplify p=x+x+y+y
Write p = 2(x+y) as p=2x+2y
Give pupils three sets of cards: the first with
formulae in words, the second with the same
formulae but expressed algebraically, the third
with a range of calculations that match the
formulae (more than one for each). Ask them to
sort the cards. Formulae should involve up to
two operations, with some including brackets.
Give pupils examples of multiplying out a
bracket with errors. Ask them to identify and talk
through the errors and how they should be
corrected, e.g.
 4(b +2) = 4b + 2
 3(p - 4) = 3p - 7
 -2 (5 - b) = ‾10 -2b
 12 – (n – 3) = 9 – n
Similarly for simplifying an expression.
Can you write an expression that would simplify
to, e.g.:
6m – 3n, 8(3x + 6)?
Are there others?
Can you give me an expression that is
equivalent to, e.g.
4p + 3q - 2?
Are there others?
What do you look for when you have an
expression to simplify? What are the important
stages?
What hints and tips would you give to someone
about simplifying expressions?…removing a
bracket from an expression?
When you substitute a = 2 and b = 7 into the
formula t = ab + 2a you get 18. Can you make
up some more formulae that also give t = 18
when a = 2 and b = 7 are substituted?
How do you go about linking a formula
expressed in words to a formulae expressed
algebraically?
Could this formulae be expressed in a different
way, but still be the same?
Use formulae from mathematics and other subjects;
Level 7
substitute numbers into expressions and formulae;
derive a formula and, in simple cases, change its subject
Find the value of these expressions
Given a list of formulae ask: If you are
3x2 + 4
4x3 – 2x
substituting a negative value for the variable,
When x = -3, and when x = 0.1
which of these might be tricky? Why?
Make l or w the subject of the formula
P = 2(l + w)
Make C the subject of the formula
F = 9C/5 + 32
Make r the subject of the formula
A = πr2
Talk me through the steps involved in this
formula. How do you know you do … before …
when substituting values into this formula?
What are the similarities and differences
between rearranging a formula and solving an
equation?
Derive and use more complex formulae and change
Level 8
the subject of a formula
Change the subject of a formula, including cases How do you decide on the steps you need to
where the subject occurs twice such as:
take to rearrange a formula? What are the
y – a = 2(a +7).
important conventions?
What strategies do you use to rearrange a
By formulating the area of the shape below in
formula where the subject occurs twice?
different ways, show that the expressions
a2 - b2 and (a - b)(a + b) are equivalent.
Talk me through how you went about deriving
this formula.
Derive formulae such as
Evaluate algebraic formulae, substituting fractions,
decimals and negative numbers
Substitute integers and fractions into formulae,
Given a list of formulae ask: If you are
including formulae with brackets, powers and
substituting a negative value for the variable,
roots. For example:
which of these might be tricky? Why?
2
2
p = s + t + (5√(s + t ))/3
Work out the value of p when s = 1.7 and t = 0.9 Talk me through the steps involved in evaluating
this formula. What tells you the order of the
steps?
From a given set of algebraic formulae, select
the examples that you typically find easy/difficult.
What makes them easy/difficult?
Manipulate algebraic formulae, equations and expressions,
finding common factors and multiplying two linear expressions
Solve linear equations involving compound
Give pupils examples of the steps towards the
algebraic fractions with positive integer
solution of equations with typical mistakes in
denominators, e.g.
them. Ask them to pinpoint the mistakes and
(2x – 6) – (7 – x) = -6
explain how to correct.
4
2
Simplify:
Talk me through the steps involved in
(2x – 3)(x – 2)
simplifying this expression. What tells you the
10 – (15 – x)
order of the steps?
(3m – 2)2 - (1 - 3m)2
How do you go about finding common factors
in algebraic fractions?
Cancel common factors in a rational expression
Give me three examples of algebraic fractions
4( x  3) 2
such as
that can be cancelled and three that cannot be
( x  3)
cancelled. How did you do it?
Expand the following, giving your answer in the
simplest form possible:
How is the product of two linear expressions of
(2b - 3)2
the form (2a ± b) different from (a ± b)?
Solving Inequalities
Solve inequalities in one variable and represent
Level 7
the solution set on a number line
An integer n satisfies -8 < 2n ≤ 10.
How do you go about finding the solution set for
List all possible values of n.
an inequality?
Solve these inequalities marking the solution set
on a number line
 3n+4 < 17 and n >2
 2(x – 5) ≤ 0 and x >-2
What are the important conventions when
representing the solution set on a number line?
Why does the inequality sign change when you
multiply or divide the inequality by a negative
number?
Solve inequalities in two variables and find the solution set
Level 8
What are the similarities and differences
between solving a pair of simultaneous
equations and solving inequalities in two
variables?
Convince me that you need a minimum of three
linear inequalities to describe a closed region.
How do you check if a point lies
 inside the region
 outside the region
 on the boundary of the region?
Manipulating Expressions
Construct, express in symbolic form, and use simple
Level 5
formulae involving one or two operations
Use letter symbols to represent unknowns and
How do you know if a letter symbol represents
variables.
an unknown or a variable?
Understand that letter symbols used in algebra
stand for unknown numbers or variables and not What are the important steps when substituting
labels, e.g. ‘5a’ cannot mean ‘5 apples’
values into this expression/formula?
What would you do first? Why?
Know and use the order of operations and
How would you continue to find the answer?
understand that algebraic operations follow the
same conventions as arithmetic operations
How are these two expressions different?
Recognise that in the expression 2 + 5a the
multiplication is to be performed first
Understand the difference between expressions
such as:
 2n and n +2
 3(c + 5) and 3c + 5
 n² and 2n
 2n² and (2n)²
Simplify or transform linear expressions by
collecting like terms; multiply a single term over
a bracket.
Simplify these expressions:
 3a + 2b + 2a – b
 4x + 7 + 3x – 3 – x
 3(x + 5)
 12 – (n – 3)
 m(n – p)
 4(a + 2b) – 2(2a + b)
Substitute integers into simple formulae, e.g.
Find the value of these expressions when a = 4.
3a2 + 4
2a3
Find the value of y when x = 3
y = 2x + 3
x
y=x-1
x +1
Simplify p=x+x+y+y
Write p = 2(x+y) as p=2x+2y
Give pupils three sets of cards: the first with
formulae in words, the second with the same
formulae but expressed algebraically, the third
with a range of calculations that match the
formulae (more than one for each). Ask them to
sort the cards. Formulae should involve up to
two operations, with some including brackets.
Give pupils examples of multiplying out a
bracket with errors. Ask them to identify and talk
through the errors and how they should be
corrected, e.g.
 4(b +2) = 4b + 2
 3(p - 4) = 3p - 7
 -2 (5 - b) = ‾10 -2b
 12 – (n – 3) = 9 – n
Similarly for simplifying an expression.
Can you write an expression that would simplify
to, e.g.:
6m – 3n, 8(3x + 6)?
Are there others?
Can you give me an expression that is
equivalent to, e.g.
4p + 3q - 2?
Are there others?
What do you look for when you have an
expression to simplify? What are the important
stages?
What hints and tips would you give to someone
about simplifying expressions?…removing a
bracket from an expression?
When you substitute a = 2 and b = 7 into the
formula t = ab + 2a you get 18. Can you make
up some more formulae that also give t = 18
when a = 2 and b = 7 are substituted?
How do you go about linking a formula
expressed in words to a formulae expressed
algebraically?
Could this formulae be expressed in a different
way, but still be the same?
Square a linear expression, and expand and simplify
Level 7
the product of two linear expressions of the form (x ± n)
and simplify the corresponding quadratic expression
Multiply out these brackets and simplify the
What is special about the two linear expressions
result:
that, when expanded, have:
(x + 4)(x - 3)
 a positive x coefficient?
(a + b)2
 a negative x coefficient?
(p - q)2
 no x coefficient?
(3x + 2)2
(a + b)(a - b)
How did you multiply out the brackets?
Show me an expression in the form (x + a)(x + b)
which when expanded has:
(i) the x coefficient equal to the constant term
(ii) the x coefficient greater than the constant
term.
What does the sign of the constant term tell you
about the original expression?
Factorise quadratic expressions including the difference of two squares
Level 8
Factorise
When reading a quadratic expression that you
x2 + 5x + 6
need to factorise, what information is critical for
x2 + x - 6
working out the two linear factors?
x2 + 5x
What difference does it make if the constant term
Recognise that
is negative?
x2 – 9 = (x + 3)(x – 3)
What difference does it make if the constant term
is zero?
Talk me through the steps you take when
factorising a quadratic expression.
Show me an expression which can be written as
the difference of two squares. How can you tell?
Why must 1000 x 998 give the same result as
9992 -1?
Manipulate algebraic formulae, equations and expressions, finding common factors and multiplying
two linear expressions
Solve linear equations involving compound
Give pupils examples of the steps towards the
algebraic fractions with positive integer
solution of equations with typical mistakes in
denominators, e.g.
them. Ask them to pinpoint the mistakes and
(2x – 6) – (7 – x) = -6
explain how to correct.
4
2
Simplify:
Talk me through the steps involved in simplifying
(2x – 3)(x – 2)
this expression. What tells you the order of the
10 – (15 – x)
steps?
2
2
(3m – 2) - (1 - 3m)
How do you go about finding common factors in
algebraic fractions?
Cancel common factors in a rational expression
Give me three examples of algebraic fractions
4( x  3) 2
such as
that can be cancelled and three that cannot be
( x  3)
cancelled. How did you do it?
Expand the following, giving your answer in the
simplest form possible:
How is the product of two linear expressions of
(2b - 3)2
the form (2a ± b) different from (a ± b)?
Sequences
Recognise and describe number patterns
Describe sequences in words, for example:
8, 16, 24, 32, 40, …
5, 13, 21, 29, 37, …
89, 80, 71, 62, 53, …
Continue sequences including those involving
decimals.
Level 4
What do you notice about this sequence of
numbers?
Can you describe a number sequence to me so,
without showing it to me, I could write down the
first ten numbers?
How do you go about finding missing numbers in
Find missing numbers in sequences.
a sequence?
Recognise and describe number relationships including multiple, factor and square
Use the multiples of 4 to work out the multiples
Which numbers less than 100 have exactly three
of 8.
factors?
Identify factors of two-digit numbers
What number up to 100 has the most factors?
Know simple tests for divisibility for 2, 3, 4, 5, 6,
8, 9
The sum of four even numbers is a multiple of
four. When is this statement true? When is it
false?
Find the factors of a number by checking for
divisibility by primes. For example, to find the
factors of 123 check mentally or otherwise, if the
number divides by 2, then 3,5,7,11…
Can a prime number be a multiple of 4? Why?
Can you give me an example of a number
greater than 500 that is divisible by 3? How do
you know?
How do you know if a number is divisible by 6?
etc
Can you give me an example of a number
greater that 100 that is divisible by 5 and also by
3? How do you know?
Is there a quick way to check if a number is
divisible by 25?
Recognise and use number patterns and relationships
Level 5
Find:
Talk me through an easy way to do this
 A prime number greater than 100
multiplication/division mentally. Why is
 The largest cube smaller than 1000
knowledge of factors important for this?
 Two prime numbers that add to 98
How do you go about identifying the factors of a
Give reasons why none of the following are
number greater than 100?
prime numbers:
4094, 1235, 5121
What is the same / different about these
sequences:
Use factors, when appropriate, to calculate
 4.3, 4.6, 4.9, 5.2, …
mentally, e.g.
 16.8, 17.1, 17.4, 17.7, …
35 × 12 = 35 × 2 × 6
 9.4, 9.1, 8.8, 8.5, ...
Continue these sequences:
 8, 15, 22, 29, …
 6, 2, -2, -6, …
 1, 1⁄2, 1⁄4, 1⁄8,
 1, -2, 4, -8
 1, 0.5, 0.25
 1, 1, 2, 3, 5, 8
I’ve got a number sequence in my head. How
many questions would you need to ask me to be
sure you know my number sequence? What are
the questions?
Solving Equations
Use systematic trial and improvement methods and ICT tools
Level 6
to find approximate solutions to equations such as x3 + x = 20
Use systematic trial and improvement methods
How do you go about choosing a value (of x) to
to find approximate solutions to equations. For
start?
example:
How do you use the previous outcomes to
 a3+ a = 20
decide what to try next?
 y(y + 2) = 67.89
Use trial and improvement for equivalent
problems, e.g.
 A number plus its cube is 20, what’s the
number?
 The length of a rectangle is 2cm longer than
the width. The area is 67.89cm2. What’s the
width?
Pupils should have opportunities to use a
spreadsheet for trial and improvement methods
How do you know when to stop?
How would you improve the accuracy of your
solution?
Is your solution exact?
Can this equation be solved using any other
method? Why?
Construct and solve linear equations with integer coefficients, using an appropriate method
Solve, e.g.
How do you decide where to start when solving
 3c – 7 = -13
a linear equation?
 4(z + 5) = 84
 4(b – 1) – 5(b + 1) = 0
Given a list of linear equations ask:
 12 / (x+1) = 21 / (x + 4)
Which of these are easy to solve?
Which are difficult and why?
Construct linear equations.
What strategies are important with the difficult
For example:
ones?
 The length of a rectangle is three times
6 = 2p – 8. How many solutions does this
its width.
equation have? Give me other equations with
Its perimeter is 24cm.
the same solution? Why do they have the same
Find its area.
solution? How do you know?
How do you go about constructing equations
from information given in a problem? How do
you check whether the equation works?
Use algebraic and graphical methods to solve simultaneous
linear equations in two variables
Given that x and y satisfy the equation 5x + y =
49 and y= 2x, find the value of x and y using an
algebraic method
Solve these simultaneous equations using an
algebraic method: 3a + 2b = 16, 5a - b=18
Solve graphically the simultaneous equations
x + 3y = 11 and 5x - 2y = 4
Level 7
What methods do you use when solving a pair of
simultaneous linear equations? What helps you
to decide which method to use? Why might you
use more than one method? Talk me through a
couple of examples with your reasons for your
chosen method.
Is it possible for a pair of simultaneous equations
to have two different pairs of solutions or to have
no solution? How do you know?
How does a graphical representation help to
know more about the number of solutions?
Manipulate algebraic formulae, equations and expressions,
Level 8
finding common factors and multiplying two linear expressions
Solve linear equations involving compound
Give pupils examples of the steps towards the
algebraic fractions with positive integer
solution of equations with typical mistakes in
denominators, e.g.
them. Ask them to pinpoint the mistakes and
(2x – 6) – (7 – x) = -6
explain how to correct.
4
2
Simplify:
Talk me through the steps involved in simplifying
(2x – 3)(x – 2)
this expression. What tells you the order of the
10 – (15 – x)
steps?
2
2
(3m – 2) - (1 - 3m)
How do you go about finding common factors in
algebraic fractions?
Cancel common factors in a rational expression
Give me three examples of algebraic fractions
4( x  3) 2
such as
that can be cancelled and three that cannot be
( x  3)
cancelled. How did you do it?
Expand the following, giving your answer in the
simplest form possible:
How is the product of two linear expressions of
(2b - 3)2
the form (2a ± b) different from (a ± b)?
Averages
Understand and use the mode and range to describe sets of data
Level 4
Use mode and range to describe data relating
List a small set of data that has a mode of 5.
to, for example, shoe sizes in the pupils’ class
How did you do it?
and begin to compare their data with data from
another class.
List a small set of data that has a mode of 5
and a range of 10. How did you work this out?
Discuss questions such as:
How can we find out if this is true?
Can you find two different small sets of data
What information should we collect?
that have the same mode and range? How did
How shall we organise it?
you do it?
What does the mode tell us?
What does the range tell us?
Understand and use the mean of discrete data and compare two simple
Level 5
distributions, using the range and one of mode, median or mean
Describe and compare two sets of football
The mean height of a class is 150cm. What
results, by using the range and mode
does this tell you about the tallest and shortest
pupil?
In your class girls are taller than boys. True or
false?
Tell me how you know.
Solve problems such as, ‘Find 5 numbers where
the mode is 6 and the range is 8’
How long do pupils take to travel to school?
Compare the median and range of the times
taken to travel to school for two groups of pupils
such as those who travel by bus and those who
travel by car.
Which newspaper is easiest to read?
In a newspaper survey of the numbers of letters
in 100-word samples the mean and the range
were compared:
 Tabloid: mean 4.3 and range 10,
 Broadsheet: mean 4.4 and range 14
Find 5 numbers that have a mean of 6 and a
range of 8. How did you do it? What if the
median was 6 and the range 8? What if the
mode was 6 and the range 8?
Two distributions both have the same range
but the first one has a median of 6 and the
second has a mode of 6. Explain how these
two distributions may differ.
Estimate the mean, median and range of a set of grouped data and
Level 7
determine the modal class, selecting the statistic most appropriate
to the line of enquiry
Why is it only possible to estimate the mean
Estimate the median and range from a grouped
(median, range) from grouped data?
frequency table.
Why is the mid-point of the class interval used
Calculate an estimate of the mean from a large
to calculate an estimated mean? Why not the
set of grouped data.
end of the class interval?
Appreciate a distinction between 'estimating the
mean of...' and 'calculating an estimate of the
mean of...'.
Produce sets of grouped data with:
1. an estimated range of 35
2. an estimated median of 22.5
3. an estimated median of 22.5 and an
estimated range of 35
4. an estimated mean of 7.4 (to 1dp)
5. …..
Talk me through the steps you take to estimate
the median from grouped data.
Why is it important to use the lowest class
value for the first class and the highest class
value for the last class to estimate range? Why
not the mid-point?
Can you construct a spreadsheet of grouped
data to calculate an estimated mean?
When is the estimated mean not the most
appropriate average, i.e. it is misleading as a
representative value.
.
Compare two or more distributions and make inferences,
using the shape of the distributions and measures of average and range
Use frequency diagrams to compare two
Make some statements from the shape of this
distributions. Calculate the mean, median and
distribution.
mode for the same data. Use shape of
distribution and measures of mean and range to, What are the key similarities and differences
for example:
between these two distributions?
 Compare athletic performances such as long
jump in Year 7 and Year 9 pupils
How would you decide whether to, for example
‘buy brand A’ instead of ‘brand B’.
 Compare the ages of the populations of the
UK and Brazil
Collecting and Recording Data
Collect and record discrete data.
Record discrete data using a frequency table
Structure data into a frequency table for
enquiries such as:
 the number of goals scored during one
season by a hockey team
Level 4
How did you decide on how to structure your
table of results?
Why did you choose these items? Might there
be others?
How did you go about collecting the data for this
enquiry?
What made the information easy or difficult to
record?
Group data, where appropriate, in equal class intervals
Decide on a suitable class interval when
Why did you choose this group size for
collecting or representing data, for example:
organising the data? What would happen if you
chose a different group size?
 pupils’ time per week spent watching
How do you know these are equal class
television – using intervals of one hour,
intervals?
 how long pupils take to travel to school Why is it important to use equal class intervals?
using intervals of 5 minutes
Ask questions, plan how to answer them and
collect the data required
Respond to given problems by asking related
questions. For example:
Problem
A neighbour tells you that the local bus service is
not as good as it used to be.
How could you find out if this is true?
Related questions
How can ‘good’ be defined? Frequency of
service, cost of journey, time taken, factors
relating to comfort, access…?
How does the frequency of the bus service vary
throughout the day/week?
Decide which data would be relevant to the
enquiry and possible sources.
Relevant data might be obtained from:
• a survey of a sample of people;
• an experiment involving observation, counting
or measuring;
• secondary sources such as tables, charts or
graphs, from reference books, newspapers,
websites, CD-ROMs and so on.
Examples of questions that pupils might explore:
 How do pupils travel to school?
 Do different types of newspaper use words
(or sentences) of different lengths?
Level 5
What was important in the way that you chose to
collect data? How do know that you will not need
to collect any more data?
How will you make sense of the data you have
collected? What options do you have in
organising the data? What other questions
could you ask of the data?
How will you make use of the data you have
collected?
Design a survey or experiment to capture the necessary
Level 6
data from one or more sources; design, trial and if necessary
refine data collection sheets; construct tables for large
discrete and continuous sets of raw data, choosing suitable
class intervals; design and use two-way tables
Investigate jumping or throwing distances:
What was important in the design of your data
 Check that the data collection sheet is
collection sheet?
designed to record all factors that may have
a bearing on the distance jumped or thrown,
What changes did you make after trialling the
such as age or height.
data collection sheet and why?
 Decide the degree of accuracy needed for
each factor.
Why did you choose that size of sample?
 Recognise that collecting too much
information will slow down the experiment;
What decisions have you made about the
too little may limit its scope.
degree of accuracy in the data you are
collecting?
Study the distribution of grass:
How will you make sense of the data you have
 Use a quadrat or points frame to estimate
collected? What options do you have in
the number of grass and non-grass plants
organising the data, including the use of two-way
growing in equal areas at regular intervals
from a north-facing building. Repeat next to a tables?
south-facing building.
How did you go about choosing your class
 Increase accuracy by taking two or more
intervals? Would different class intervals make
independent measurements.
a difference to the findings? How?
Suggest a problem to explore using statistical methods,
Level 7
frame questions and raise conjectures; identify possible
sources of bias and plan how to minimise it
Decide on questions to explore in a given
What convinced you that this question could
context and raise conjectures, e.g. in PSHE
usefully be explored using statistical methods?
questions posed might be:
 How available are fairly-traded goods in local Convince me that it is valid to explore this
situation / hypothesis.
shops?
 Who buys fairly-traded goods?
How can you tell if data is biased? Can you give
A conjecture might be:
any examples of where data may be biased?
People with experience or links with LEDCs are
more likely to be aware of and to buy fairlytraded goods.
How would you go about reducing bias? Any
general guidelines for checking for bias?
Be aware of bias and have strategies to reduce
bias, for example due to selection, non-response Data is biased. Always, never or sometimes
true?
or timing.
.
Probability



In probability, select methods based on equally likely
Level 5
outcomes and experimental evidence, as appropriate
Find and justify probabilities based on equally likely
Can you give me an example of an event for which
outcomes in simple contexts. For example:
the probability can only be calculated through an
The letters of the word REINDEER are written on experiment?
8 cards, and a card is chosen at random. What is the
Can you give me an example of what is meant by
probability that the chosen letter is an E?
‘equally likely outcomes’?
On a fair die what is the probability of rolling a
prime number?
Estimate probabilities from experimental data. For
example:
Test a dice or spinner and calculate probabilities
based on the relative frequency of each score.
Decide if a probability can be calculated or if it can
only be estimated from the results of an experiment.
Understand and use the probability scale from 0 to 1
What words would you use to describe an event with What is the same / different with a probability scale
a probability of 90%? What about a probability of
marked with:
0.2? Sketch a probability scale, and mark these

fractions
probabilities on it.

decimals

percentages

words?




See pages 278 and 280 of the Framework
supplement of examples


Give examples of probabilities (as percentages) for
events that could be described using the following
words:
Impossible
Almost (but not quite) certain
Fairly likely
An even chance
Make up examples of any situation with equally
likely outcomes with given probabilities of: 0.5, 1/6,
0.2 etc. Justify your answers.
Understand that different outcomes may result from repeating an experiment
Understand that if an experiment is repeated there
'When you spin a coin, the probability of getting a
may be, and usually will be, different outcomes. For
head is 0.5. So if you spin a coin ten times you
example:
would get exactly 5 heads'. Is this statement true or
false? Why?
Compare estimated probabilities obtained by
testing a piece of apparatus (such as a dice, spinner What is wrong with this coin!?’ What do you think?
or coin) with those obtained by other groups.
You spin a coin a hundred times and count the
number of times you get a head. A robot is
Understand that increasing the number of times an
programmed to spin a coin a thousand times. Who
experiment is repeated generally leads to better
is most likely to be closer to getting an equal number
estimates of probability. For example:
Confirm that the experimental probabilities for the of heads and tails? Why?
scores on an ordinary dice approach the theoretical
values as the number of trials increase.
Find and record all possible mutually exclusive outcomes
for single events and two successive events in a systematic way
Use a sample space diagram to show all outcomes
when two dice are thrown together, and the scores
added.
One red and one white dice are both numbered 1 to
6. Both dice are thrown and the scores added. Use
a sample space to show all possible outcomes.



Level 6
Give me examples of mutually exclusive events.
How do you go about identifying all the mutually
exclusive outcomes for an experiment?
What strategies do you use to make sure you have
found all possible mutually exclusive outcomes for
two successive events, for example rolling two dice?
How do you know you have recorded all the possible
outcomes?
Know that the sum of probabilities of all mutually exclusive outcomes is 1 and use this when solving
problems
Two coins are thrown at the same time. There are
four possible outcomes: HH, HT, TH, and TT.
The probability of throwing two heads is ¼. What is
Why is it helpful to know that the sum of probabilities
the probability of not throwing two heads?
of all mutually exclusive events is 1? Give me an
Extend this to:
example of how you have used this when working
Three coins
on probability problems.
Four coins
Five coins.
Understand relative frequency as an estimate of probability
Level 7
and use this to compare outcomes of an experiment
Recognise that repeated trials result in experimental What might be different about using theoretical
probability tending to a limit, and that this limit may
probability to find the probability of obtaining a 6
be the only way to estimate probability. Describe
when you roll a dice, and using experimental
situations where the use of experimental data to
probability for the same purpose?
estimate a probability would be necessary.
True or false

Experimental probability is more reliable than
theoretical probability;

Experimental probability gets closer to the true
probability as more trials are carried out;

Relative frequency finds the true probability.
Know when to add or multiply two probabilities
Recognise when probabilities can be associated with
independent events or mutually exclusive.


Understand that when A and B are mutually

exclusive, then the probability of A or B occurring is
P(A) + P(B), whereas if A and B are independent
events, the probability of A and B occurring is P(A) ×
P(B)
Solve problems such as:
A pack of blue cards are numbered 1 to 10. What
is the probability of picking a multiple of three or a
multiple of 5?
There is also an identical pack of red cards. What is
the probability of picking a red 5 and a blue 7?
Level 8
Show me an example of:
A problem which could be solved by adding
probabilities
A problem which could be solved by multiplying
probabilities
What are all the mutually exclusive events for this
situation? How do you know they are mutually
exclusive? Why do you add the probabilities to find
the probability either this event or this event
occurring?
If I throw a coin and roll a dice the probability of a 5
and a head is 1/12. This is not 1/2 + 1/6. Why not?
What are the mutually exclusive events in this
problem? How would you use these to find the
probability?
Use tree diagrams to calculate probabilities of combinations of independent events
Construct tree diagrams for a sequence of events
How can you distinguish between mutually exclusive
using given probabilities or theoretical probabilities.
and independent events from a tree diagram?
Use the tree diagram to calculate probabilities of
combinations of independent events.
Why do the probabilities on each pair of branches
have to sum to 1?
The probability that it will rain on Tuesday is 1/5.
The probability that it will rain on Wednesday is 1/3.
How can you tell from a completed tree diagram
What is the probability that it will rain on just one of
whether the question specified with or without
the days?
replacement?
The probability that Nora fails her driving theory test
on the first attempt is 0.1. The probability that she
passes her practical test on the first attempt is 0.6.
Complete a tree diagram based on this information
and use it to find the probability that she passes both
tests on the first attempt.
What strategies do you use to check the
probabilities on your tree diagram are correct?
Talk me through the steps you took to construct this
tree diagram and use it to find the probability of this
event.
Representing and Interpreting Data
Continue to use Venn and Carroll diagrams to record
Level 4
their sorting and classifying of information
Using this Carroll diagram for numbers, write a
Give me an example of a Venn diagram that can
number less than 100 in each space
be used to sort the numbers 1-50. Which criteria
have you used and why?
even
not even
a square
number
not a
square
number
Use a Venn diagram to sort by two criteria, e.g.
sorting numbers using the properties ‘multiples
of 8’ and ‘multiples of 6’
Give me an example of a Carroll diagram – with
four cells – that can be used to sort the numbers
1-50
Which criteria have you used and why?
What are the important steps when completing a
Carroll diagram? What strategies do you use to
check your Carroll diagram is complete?
What are the important steps when completing a
Venn diagram? What strategies do you use to
check your Venn diagram is complete?
How is a Venn diagram different from a Carroll
diagram?
Construct and interpret frequency diagrams and simple line graphs
Suggest an appropriate frequency diagram to
For a given graph/table/chart, make up three
represent particular data, for example decide
questions that can be answered using the
whether a bar chart, Venn diagram or pictogram information represented.
would be most appropriate. For pictograms use
one symbol to represent several units
What makes the information easy or difficult to
represent?
Decide upon an appropriate scale for a graph
e.g. labelled divisions representing 2, 5, 10, 100 How do you decide on the scale to use on the
vertical axis? How would a different scale
Interpret simple pie charts
change the graph?
Interpret the scale on bar charts and line graphs,
reading between the labelled divisions e.g.
reading 17 on a scale labelled in fives
Interpret the total amount of data represented,
compare data sets and respond to questions
e.g. how does our data about favourite television
programmes compare to the data from year 8
pupils?
Interpret graphs and diagrams, including pie charts,
Level 5
and draw conclusions
Interpret, by comparing the cells in a 2-way
Make up a statement or question for this
table,
chart/graph using one or more of the following
key words:
Interpret bar charts with grouped data
 total, range, mode;
 fraction, percentage, proportion.
Interpret and compare pie charts where it is not
necessary to measure angles
Read between labelled divisions on a scale of a
graph or chart, for example read 34 on a scale
labelled in tens or 3.7 on a scale labelled in
ones, and find differences to answer, ‘How much
more…?’
Recognise when information is presented in a
misleading way, for example compare two pie
charts where the sample sizes are different
When drawing conclusions, identify further
questions to ask.
Create and interpret line graphs where the intermediate values have meaning
Draw and use a conversion graph for Pounds
Do the intermediate values have any meaning
and Euros
on these graphs? How do you know?
 Show graphs where there is no meaning –
Answer questions based on a graph showing
for example, a line graph showing the trend
tide levels; for example,
in midday temperatures over a week.
 Between which times will the height of the
 Also show examples where interpolation
tide be greater than five metres?
makes sense – for example, the temperature
in a classroom, measured every 30 minutes
Use a line graph showing average maximum
for six hours.
monthly temperatures for two locations. For
example:
Convince me that you can use this graph
 What was the coldest month in Manchester? (conversion graph between litres and gallons –
up as far as 20 gallons) to find out how many
 During which months would you expect the
litres are roughly equivalent to 75 gallons
maximum temperatures in Manchester and
Sydney to be about the same?
Select, construct and modify, on paper & using ICT:
Level 6
 pie charts for categorical data;
 bar charts and frequency diagrams for discrete and continuous data;
 simple time graphs for time series;
 scatter graphs.
and identify which are most useful in the context of the problem
Understand that pie charts are mainly suitable
for categorical data. Draw pie charts using ICT
When drawing a pie chart, what information do
and by hand, usually using a calculator to find
you need to calculate the size of the angle for
angles
each category?
Draw compound bar charts with subcategories
Use frequency diagrams for continuous data and
know that the divisions between bars should be
labelled
What is discrete/continuous data? Give me
some examples.
How do you go about choosing class intervals
when grouping data for a bar chart/frequency
diagram?
Use line graphs to compare two sets of data.
What’s important when choosing the scale for
the frequency axis?
Use scatter graphs for continuous data, two
variables, showing, for example, weekly hours
worked against hours of TV watched.
Is this graphical representation helpful in
addressing the hypothesis? If not, why and what
would you change?
When considering a range of graphs
representing the same data:
 which is the easiest to interpret? Why?
 Which is most helpful in addressing the
hypothesis? Why?
Communicate interpretations and results of a statistical survey using selected tables, graphs
and diagrams in support
Using selected tables, graphs and diagrams for
Which of your tables/graphs/diagrams give the
support; describe the current incidence of male
strongest evidence to support/reject your
and female smoking in the UK, using frequency
hypothesis? How?
diagrams to demonstrate the peak age groups.
Show how the position has changed over the
What conclusions can you draw from your
past 20 years; using line graphs. Conclude that
table/graph/diagram?
the only group of smokers on the increase is
females aged 15 -25.
Convince me using the table/graph/diagram.
What difference would it make if this piece of
data was included?
Are any of your graphs/diagrams difficult to
interpret? Why?
Select, construct and modify, on paper and using ICT,
Level 7
suitable graphical representation to progress an enquiry,
including frequency polygons and lines of best fit on scatter graphs
Use superimposed frequency polygons to
How does your graph help with your statistical
compare results.
analysis? What does it show? What are the
benefits and any disadvantages of your chosen
graph? Did you try any graphs that were not
When plotting a line of best fit, find the mean
helpful? Why?
point and make predictions using a line of best
fit.
Convince me that this is the most appropriate
graph to use in this case.
Recognise that a prediction based on a line of
best fit may be subject to error.
How do you go about labelling the horizontal
axis when constructing a frequency polygon?
Recognise the potential problems of extending
the line of best fit beyond the range of known
values.
How can you tell if it is sensible to draw a line of
best fit on a scatter graph?
Examine critically the results of a statistical enquiry,
and justify the choice of statistical representation in written presentation
Examine data for cause and effect
Can you think of an example where there is
positive correlation, but it is unlikely that there is
causation?
Analyse data and try to explain anomalies. For
example: In a study of engine size and
acceleration times, observe that in general a
Which of the following statements is more
larger engine size leads to greater acceleration.
precise: 'my hypothesis is true' or 'there is strong
However, particular cars do not fit the overall
evidence to support my hypothesis'. Why?
pattern.
Reasons may include because they are heavier
Convince me that there is evidence to support
than average or are built for rough terrain rather
your hypothesis.
than normal roads.
Why might it not be the case that:
 Your hypothesis is true if you have found
Recognise that establishing a correlation or
enough evidence to support it
connection in statistical situations does not
 You have failed if your hypothesis appears to
necessarily imply that one variable causes
be flawed.
change to another (i.e. ‘correlation does not
imply causality’), and that there may be external Convince me that you have chosen the most
factors affecting both.
appropriate statistical measure to use in this
case. What else did you consider and not use?
Compare two or more distributions and make inferences,
Level 8
using the shape of the distributions and measures of average
and spread including median and quartiles
Construct, interpret and compare box-plots for
What features of the distributions can you
two sets of data, for example, the heights of
compare when using a box plot?…a frequency
Year 7 and Year 9 pupils or the times that Year
diagram?
9 boys and Year 9 girls can run 100m.
Make some comparative statements from the
shape of each of these distributions.
Recognise positive and negative skew from the
shape of distributions represented in:
What are the key similarities and differences
between these two distributions?
 frequency diagrams
 cumulative frequency diagrams
Describe two contexts, one in which a
 box plots
variable/attribute has negative skew and the
other in which it has positive skew?
How can we tell from a box plot that the variable
has negative skew?
Convince me that the following representations
are from different distributions, for example:
What would you expect to be the same/different
about the two distributions representing, for
example
 Heights of pupils in Year 1 to 6 and Heights
of pupils only in Year 6
Directed Numbers
Round decimals to the nearest decimal place
Level 5
and order negative numbers in context
Petrol costs 124.9p a litre. How much is this Explain whether the following are true or
to the nearest penny?
false:
 2.399 rounds to 2.310 to 2 decimal
Round, e.g.
places
 2.75 to 1 decimal place
 -6 is less than -4
 176.05 to 1 decimal place
 3.5 is closer to 4 than it is to 3
 25.03 to 1 decimal place
 -36 is greater than -34
 24.992 to 2 decimal places
 8.4999 rounds to 8.5 to 1 decimal place
Order the following places from coldest to
warmest:
 Moscow, Russia: 4C
 Oymyakou, Russia: -96C
 Vostok, Antarctica: -129C
 Rogers Pass, Montana, USA: -70C
 Fort Selkirk, Yukon, Canada: -74C
 Northice, Greenland: -87C
 Reykjavik, Iceland: 5C
How do you go about rounding a number to
one decimal place?
Why might it not be possible to identify the
first three places in a long jump competition if
measurements were taken in metres to one
decimal place?
Show me a length that rounds to 4.3m to one
decimal place. Are there other lengths?
What is the same / different about these
numbers:
72.344 and 72.346
Estimation
Check the reasonableness of results with reference
Level 4
to the context or size of numbers
Use rounding to approximate and judges
Roughly what answer do you expect to get?
whether the answer is the right order of
How did you come to that estimate?
magnitude, for example:
2605- 1897 is about 3000-2000
Do you expect your answer to be less than or
12% of 192 is about 10% of 200
greater than your estimate? Why?
Discuss questions such as:
 A girl worked out the cost of 8 bags of apples
at 47p a bag. Her answer was £4.06.
Without working out the answer, say whether
you think it is right or wrong.
 A boy worked out how many 19p stamps you
can buy for £5. His answer was 25. Do you
think he was right or wrong? Why?
 A boy worked out £2.38 + 76p on a
calculator. The display showed 78.38. Why
did the calculator give the wrong answer?
Round decimals to the nearest decimal place
and order negative numbers in context
Petrol costs 124.9p a litre. How much is this to
the nearest penny?
Round, e.g.
 2.75 to 1 decimal place
 176.05 to 1 decimal place
 25.03 to 1 decimal place
 24.992 to 2 decimal places
Order the following places from coldest to
warmest:
 Moscow, Russia: 4C
 Oymyakou, Russia: -96C
 Vostok, Antarctica: -129C
 Rogers Pass, Montana, USA: -70C
 Fort Selkirk, Yukon, Canada: -74C
 Northice, Greenland: -87C
 Reykjavik, Iceland: 5C
Level 5
Explain whether the following are true or false:
 2.399 rounds to 2.310 to 2 decimal places
 -6 is less than -4
 3.5 is closer to 4 than it is to 3
 -36 is greater than -34
 8.4999 rounds to 8.5 to 1 decimal place
How do you go about rounding a number to one
decimal place?
Why might it not be possible to identify the first
three places in a long jump competition if
measurements were taken in metres to one
decimal place?
Show me a length that rounds to 4.3m to one
decimal place. Are there other lengths?
What is the same / different about these
numbers:
72.344 and 72.346
Make and justify estimates and approximations of calculations;
Level 7
estimate calculations by rounding numbers to one significant figure
and multiplying and dividing mentally
Examples of what pupils should know and be
Probing questions
able to do
Estimate answers to:
Talk me through the steps you would take to find
5.16 x
an estimate for the answer to this calculation?
0.0721 x
(186.3 x 88.6)/(27.2 x 22.8)
Would you expect your estimated answer to be
greater or less than the exact answer? How can
you tell? Can you make up an example for
which it would be difficult to decide?
Show me examples of multiplication and division
calculations using decimals that approximate to
60.
Why is 6 ÷ 2 a better approximation for 6.59 ÷
2.47 than 7 ÷ 2?
Why it is useful to be able to estimate the answer
to complex calculations?
Four Rules
Use a range of mental methods of computation with all operations
Level 4
Calculate mentally a difference such as 8006 Which of these calculations can you do without
2993 by 'counting up' or by considering the
writing anything down? Why is it sensible to
equivalent calculation of 8006 - 3000 + 7
work this out mentally? What clues did you look
for?
Work out mentally that:
4005 - 1997 = 2008 because it is
How did you find the difference? Talk me
4005 - 2000 + 3 = 2005 + 3 = 2008
through your method.
Work out mentally by counting up from the
smaller to the larger number:
Give me a different calculation that has the same
8000 - 2785 is 5 + 10 + 200 + 5000 = 5215
answer…an answer that is ten times bigger…etc.
How did you do it?
Calculate complements to 1000
Recall multiplication facts up to 10 × 10 and quickly derive corresponding division facts
Use their knowledge of tables and place value in If someone has forgotten the 8 times table, what
calculations with multiples of 10 such as 180 ÷ 3 tips would you give them to work it out? What
other links between tables are useful?
Respond rapidly to oral and written questions
like:
If you know that 4 x 7 = 28, what else do you
 Nine eights
know?
 How many sevens in 35?
 8 times 8
Start from a two-digit number with at least 6
 6 multiplied by 7
factors, e.g. 56. How many different
 Multiply 11 by 8
multiplication and division facts can you make
 7 multiplied by 0
using what you know about 56? How have you
identified the divisions?
Respond quickly to questions like
 Divide 3.6 by 9
The product is 40. Make up some questions.
 What is 88 shared between 8?
How are these different questions linked?
 0.6 times 7 times2
Know by heart or derive quickly
 Doubles of two-digit whole numbers or
decimals
 Doubles of multiples of 10 up to 1000
 Doubles of multiples of 100 up to 10 000
 And all the corresponding halves
The quotient is 5. Make up some questions.
How did you go about devising these questions?
Use efficient written methods of addition and subtraction and of short multiplication and
division
Calculate 1202 + 45 + 367 or 1025 - 336
Give pupils some examples of work with errors
for them to check, for example:
Work with numbers to two decimal places,
including sums and differences with different
numbers of digits, and totals of more than two
12.3 +
numbers, e.g.
9.8
 671.7 - 60.2
21.11;
 543.65 + 45.8
 1040.6 - 89.09
4.07
 764.78 - 56.4
-1.5
 76.56 + 312.2 + 5.07
3.57;
Use, for example, the grid method before
moving on to short multiplication.
Use efficient methods of repeated subtraction,
by subtracting multiples of the divisor, before
moving to short division
36.2
×
8
288.16
Which are correct/incorrect? How do you know?
Explain what has been done wrong and correct
the answers.
Use a calculator efficiently and appropriately to perform
Level 7
complex calculations with numbers of any size, knowing not
to round during intermediate steps of a calculation
How do you decide on the order of operations for
a complex calculation when using a calculator?
Use a calculator to evaluate more complex
calculations such as those with nested brackets
or where the memory function could be used.
Which calculator keys and functions are
For example:
important when doing complex calculations?
(Explore brackets, memories, +/- key, reciprocal
key.)
• Use a calculator to work out:
a. 45.65 × 76.8
1.05 × (6.4 – 3.8)
Give pupils some calculations with answers
(some correct but some with common mistakes)
and ask them to decide which are correct. Ask
b. 4.6 + (5.7 – (11.6 × 9.1))
them to analyse the mistakes in the ones that are
incorrect and explain the correct method using a
c. {(4.5)2 + (7.5 – 0.46)}2
calculator.
2
2
d. 5 × √(4.5 + 6 )
Why shouldn’t you round during the intermediate
3
steps of calculations?
Fractions, Decimals and Percentages
Recognise approximate proportions of a whole and
Level 4
use simple fractions and percentages to describe these
Recognise simple equivalence between
What fractions/percentages can you easily work
fractions, decimals and percentages e.g. 1/2,
out in your head? Talk me through a couple of
1/4, 1/10, 3/4
examples.
Convert mixed numbers to improper fractions
and vice versa
Express a smaller number as a fraction of a
larger one. For example:
What fraction of:
1 metre is 35 centimetres
1 kilogram is 24 grams
1 hour is 33 minutes?
Talk me through how you know that, e.g. 90 is ¾
of 120.
Approximately what fraction of, e.g. this shape is
shaded? Would you say it is more or less than
this fraction? How do you know?
When calculating percentages of quantities, what
percentage do you usually start from? How do
you use this percentage to work out others?
To calculate 10% of a quantity, you divide it by
10. So to find 20%, you must divide by 20. What
is wrong with this statement?
Using a 1 - 100 grid, 50% of the numbers are
even. How would you check? Give me a
question with the answer 20% (or other simple
percentages and fractions)
Multiply a simple decimal by a single digit
Calculate:
 2.4 x 7
 4.6 x 8
 9.3 x 9
What would you estimate the answer to be? Is
the accurate answer bigger or smaller than your
estimate? Why?
How would you help someone to understand
that, e.g. 0.4 × 7 = 2.8, 2.4 × 7 = 16.8?
Which of these calculations are easy to work out
in your head, and why?
 0.5 x 5
 0.5 x 8
 1.5 x 5
 11.5 x 8
Talk me through your method.
Order decimals to three decimal places
Place these numbers in order of size, starting
with the greatest: 0.206, 0.026, 0.602, 0.620,
0.062
Place these numbers on a line from 6.9 to 7.1:
6.93, 6.91, 6.99, 7.01, 7.06
Put these in order, largest/smallest first: 1.5,
1.375, 1.4, 1.3, 1.35, 1.425
Put these in order, largest/smallest first: 7.765,
7.675, 6.765, 7.756, 6.776
What do you look for first when you are ordering
numbers with decimals?
Which part of each number do you look at to help
you?
Which numbers are the hardest to put in order?
Why?
What do you do when numbers have the same
digit in the same place?
Give me a number between 0.12 and 0.17.
Which of the two numbers is it closer to? How do
you know?
Use equivalence between fractions and order fractions and decimals
Level 5
4
Give me two equivalent fractions. How do you
Find two fractions equivalent to
5
know they are equivalent?
Show that 12 is equivalent to 6 , 4 and 2
18
9 6
3
Give me some fractions that are equivalent to …
Find the unknown numerator or denominator in:
How did you do it?
1
7
36
= ?
= 35
= ?
4
12
?
16
48
24
Can you draw a diagram to convince me that 14 is
Write the following set of fractions in order from
smallest to largest:
1
4
, 23 , 16 , 34 , 65 , 13
the same as
line?
3
12
? Can you show me on a number
Convert fractions to decimals by using a known
equivalent fraction and using division. For
example:
 2/8 = ¼ = 0.25
 3/5 = 6/10 = 0.6
 3/8 = 0.375 using short division
Explain whether the following is true or false:
10 is greater than 9, so 0.10 is greater than 0.9
Answer questions such as:
 Which is greater, 0.23 or 3/16?
 3 , 2 , 4
Now show how you could fill in the missing
numbers in a different way, so that each set of
fractions is in descending order.
Explain how you could fill in the missing numbers
so that each resulting set of fractions is in
ascending order:
 3 , 2 , 4
What are the important steps when putting a set
of fractions/decimals in order?
Use known facts, place value, knowledge of operations and brackets to calculate including
using all four operations with decimals to two places
Explain how you would do this multiplication by
Given that 42  386 = 16 212, find the answers
using factors, e.g. 5.8 x 40
to:
 4.2  386
What clues do you look for when deciding if you
 42  3.86
can do a multiplication mentally? e.g. 5.8 x 40
 420  38.6
 16 212  0.42
Give an example of how you could use
partitioning to multiply a decimal by a two digit
Use factors to find the answers to:.
 3.2 x 30 knowing 3.2 x 10 = 32: 32 x 3 = 96 whole number, e.g. 5.3 x 23.

156 ÷ 6 knowing 156 ÷ 3 = 52
52 ÷ 2 = 26
Use partitioning for multiplication; partition either
part of the product:
7.3 x 11 = (7.3 x 10) + 7.3 = 80.3
Use 1/5 = 0.2 to convert fractions to decimals
mentally. e.g. 3/5 = 0.2 x 3 = 0.6
Calculate:
 4.2  (3.6 + 7.4)
 4.2  3.6 + 7.4
 4.2 + 3.6  7.4
 (4.2 + 3.6)  7.4
Extend doubling and halving methods to include
decimals, for example:
8.12 x 2.5 = 4.06 x 5 = 20.3
37 × 64 = 2368. Explain how you can use this
fact to devise calculations with answers 23.68,
2.368, 0.2368.
73.6 ÷ 3.2 = 23. Explain how you can use this to
devise calculations with the same answer.
Explain why the ‘standard’ compact method for
subtraction (decomposition) is not helpful for
examples like 10 008 – 59.
Talk me through this calculation. What steps do
you need to take to get the answer? How do you
know what you have to do first?
What are the important conventions for the order
of operations when doing a calculation?
Use a calculator where appropriate to calculate fractions/percentages of
quantities/measurements
Use mental strategies in simple cases, e.g.
What fractions/percentages of given quantities
can you easily work out in your head? Talk me
 1/8 of 20; find one quarter and halve the
through a couple of examples.
answer
 75% of 24; find 50% then 25% and add the
When calculating percentages of quantities, what
results
percentages do you usually start from? How do
 15% of 40; find 10% then 5% and add the
you use this percentage to work out others?
results
 40% of 400kg; find 10% then multiply by 4
How do you decide when to use a calculator,
rather than a mental or written method, when
Calculate simple fractions or percentages of a
finding fractions or percentages of quantities?
number/quantity e.g. ⅜ of 400g or 20% of £300
Give me some examples.
Use a calculator for harder examples, e.g.
Talk me through how you use a calculator to find
 181 of 207; 207  18 = 11.5
a percentage of a quantity or a fraction of a
 43% of £1.36; 0.43  1.36 = 58p
quantity
 62% of 405 m; 0.62  405 = 251.1m
Reduce a fraction to its simplest form by cancelling common factors
Cancel these fractions to their simplest form by
What clues do you look for when cancelling
looking for highest common factors:
fractions to their simplest form?
9
15
12
18
42
56
How do you know when you have the simplest
form of a fraction?
Use the equivalence of fractions, decimals and percentages to
Level 6
compare proportions
Convert fraction and decimal operators to
Which sets of equivalent fractions, decimals and
percentage operators by multiplying by 100. For percentages do you know? From one set that
example:
you know (e.g. 1/10  0.1  10%), which others
 0.45; 0.45 × 100% = 45%
can you deduce?
 7/12; (7 ÷ 12) × 100% = 58.3% (1 d.p.)
How would you go about finding the decimal and
Continue to use mental methods for finding
percentage equivalents of any fraction?
percentages of quantities
How would you find out which of these is closest
Use written methods, e.g.
to 1/3: 10/31; 20/61; 30/91; 50/151?
 Using an equivalent fraction:
13% of 48; 13/100 × 48 = 624/100 = 6.24
What links have you noticed within equivalent
 Using an equivalent decimal:
sets of fractions, decimals and percentages?
13% of 48; 0.13 × 48 = 6.24
Give me a fraction between 1/3 and 1/2. How did
you do it? Which is it closer to? How do you
know?
Calculate percentages and find the outcome of a given percentage increase or decrease
Use written methods, e.g.
Talk me through how you would
 Using an equivalent fraction: 13% of 48;
increase/decrease £12 by, for example 15%. Can
13/100 × 48 = 624/100 = 6.24
you do it in a different way? How would you find
 Using an equivalent decimal: 13% of 48;
the multiplier for different percentage
0.13 × 48 = 6.24
increases/decreases?
 Using a unitary method: 13% of 48; 1% of 48
= 0.48 so 13% of 48 = 0.48 × 13 = 6.24
The answer to a percentage increase question is
£10. Make up an easy question. Make up a
Find the outcome of a given percentage
difficult question.
increase or decrease. e.g.
 an increase of 15% on an original cost of £12
gives a new price of £12 × 1.15 = £13.80,
 or 15% of £12 = £1.80 £12 + £1.80 =
£13.80
Add and subtract fractions by writing them with a common denominator, calculate fractions
of quantities (fraction answers); multiply and divide an integer by a fraction
Add and subtract more complex fractions such
Why are equivalent fractions important when
as 11/18 + 7/24, including mixed fractions.
adding or subtracting fractions?
Solve problems involving fractions, e.g.
 In a survey of 24 pupils 1/3 liked football
best, 1/4 liked basketball, 3/8 liked athletics
and the rest liked swimming. How many
liked swimming?
What strategies do you use to find a common
denominator when adding or subtracting
fractions?
Is there only one possible common denominator?
What happens if you use a different common
denominator?
Give pupils some examples of +, - of fractions
with common mistakes in them. Ask them to talk
you through the mistakes and how they would
correct them.
How would you justify that 4 ÷ 1/5 = 20? How
would you use this to work out 4 ÷ 2/5? Do you
expect the answer to be greater or less than 20?
Why?
Understand the effects of multiplying and dividing by
numbers between 0 and 1
.
Give an approximate answer to:
357 ÷ 0.3
1099 ÷ 0.22
1476 x 0.99
57.7 ÷ 0.65
Know and understand that division by zero has
no meaning. For example, explore dividing by
successive smaller positive decimals,
approaching zero then negative decimals
approaching zero.
Recognise and use reciprocals.
Add, subtract, multiply and divide fractions
Find the area and perimeter of a rectangle
measuring 4¾ inches by 63/8 inches.
Using 22/7 as an approximation for pi, estimate
the area of a circle with diameter 28mm.
Pupils should be able to understand and use
efficient methods to add, subtract, multiply and
divide fractions, including mixed numbers and
questions that involve more than one operation.
Level 7
Multiplying makes numbers bigger. When is this
statement true and when is it false?
Division makes things smaller. When is this
statement true and when is it false?
How would you justify that dividing by ½ is the
same as multiplying by 2? What about dividing
by 1/3 and multiplying by 3? What about dividing
by 2/3 and multiplying by 3/2? How does this link
to division of fractions?




0.8 ÷ 0.1
16 × 0.5
1.6 ÷ 0.5
1.6 ÷ 0.2
Talk me through the reasoning that took you to
the answer to each of these calculations.
How do you go about adding and subtracting
more complex fractions e.g. 2 2/5 – 1 7/8?
Give me two fractions which multiply together to
give a bigger answer than either of the fractions
you are multiplying. How did you do it?
Give pupils some examples of +, - × and ÷ with
common mistakes in them (including mixed
numbers). Ask them to talk you through the
mistakes and how they would correct them.
How would you justify that dividing by ½ is the
same as multiplying by 2? What about dividing
by 1/3 and multiplying by 3? What about dividing
by 2/3 and multiplying by 3/2? How does this link
to what you know about dividing by a number
between 0 and 1?
Understand the equivalence between recurring decimals and fractions
Distinguish between fractions with denominators that have
only prime factors 2 and 5 (which are represented as
terminating decimals), and other fractions (which are
represented by terminating decimals).
Decide which of the following fractions are equivalent to
terminating decimals: 3/5, 3/11, 7/30, 9/22, 9/20, 7/16
.
Write 0. 4
.
5 as a fraction in its simplest terms
Level 8
Write some fractions which terminate when converted to a
decimal. What do you notice about these fractions? What
clues do you look for when deciding if a fraction terminates?
1/3 is a recurring decimal. What other fractions related to
one-third will also be recurring? Using the knowledge that
.
1/3 = 0. 3 How would you go about finding the decimal
equivalent of 1/6, 1/30…?
.
.
1/11 = 0. 0 9 How do you use this fact to express 2/11,
3/11, 12/11 in decimal form?
If you were to convert these decimals to fractions;
0.0454545…, 0.454545……, 4.545454……, 45.4545…..
Which of these would be easy/difficult to convert? What
makes them easy/difficult to convert?
.
Can you use the fraction equivalents of 4. 5
.
.
4
and
.
45. 4 5 to prove the second is ten times greater than the
first?
Which of the following statements are true/false:

all terminating decimals can be written as a fraction

all recurring decimals can be written as a fraction

all numbers can be written as a fraction
Use fractions or percentages to solve problems involving repeated proportional changes or
the calculation of the original quantity given the result of a proportional change
Solve problems involving, for example
Talk me through why this calculation will give the
compound interest and population growth using
solution to this repeated proportional change
multiplicative methods.
problem.
Use a spreadsheet to solve problems such as:
How long would it take to double your
investment with an interest rate of 4% per
annum?
A ball bounces to ¾ of its previous height each
bounce. It is dropped from 8m. How many
bounces will there be before it bounces to
approximately 1m above the ground?
How would the calculation be different if the
proportional change was…?
Solve problems in other contexts, for example:
Each side of a square is increased by 10%. By
what percentage is the area increased?
The length of a rectangle is increased by 15%.
The width is decreased by 5%. By what
percentage is the area changed?
Give pupils a set of problems involving repeated
proportional changes and a set of calculations.
Ask pupils to match the problems to the
calculations.
What do you look for in a problem to decide the
product that will give the correct answer?
How is compound interest different from simple
interest?
Limits
Recognise that measurements given to the nearest
Level 7
whole unit may be inaccurate by up to one half of
the unit in either direction
Suggest a range for measurements such as: Explain the difference in meaning between
123mm; 1860mm; 3.54kg; 6800m2
0.6m and 0.600m. When is it necessary to
include the zeros in measurements?
Find maximum and minimum values for a
measurement that has been rounded to a
What range of measured lengths might be
given degree of accuracy
represented by the measurement 320cm?
Solve problems such as:
The dimensions of a rectangular floor,
measured to the nearest metre, are given as
28m by 16m. What range must the area of
the floor lie within? Suggest a sensible
answer for the area, given the degree of
accuracy of the data.
What accuracy is needed to be sure a
measurement is accurate to the nearest
centimetre?
Why might you pick a runner whose time for
running 100m is recorded as 13.3 seconds
rather than 13.30 seconds? Why might you
not?
.
Explain how 6 people each weighing 110kg
might exceed a weight limit of 660kg for a lift.
Place Value
Use place value to multiply and divide whole numbers by 10 or 100
Level 4
Respond to oral and written questions such as:
Why do 6 x 100 and 60 x 10 give the same answer?
What about 30 ÷ 10 and 300 ÷ 100?
 How many times larger is 2600 than 26?
I have 37 on my calculator display. What single
 How many £10 notes are in £120, £1200? How
multiplication should I key in to change it to 3700?
many £1 coins, 10p coins, 1p coins?
Explain why this works.
 Tins of dog food at 42p each are put in packs of
10. Ten packs are put in a box. How much does Can you tell me a quick way of multiplying by 10, by
one box of dog food cost? 10 boxes? 100 boxes? 100?
Can you tell me a quick way of dividing by 10, by
Work out mentally the answers to questions such as:
100?
329 x 100 = 
8000 ÷ 100 = 
56 x  = 5600
7200 ÷  = 72
420 x  = 4200 3900 ÷  = 390
Complete statements such as:
 4 x 10 = •
 4 x • = 400
 • ÷ 10 = 40
 • x 1000 = 40 000
 • x 10 = 400
Use understanding of place value to multiply and divide whole
Level 5
numbers and decimals by 10, 100 and 1000 and explain the effect
Know for example:
How would you explain that 0.35 is greater than
 in 5.239 the digit 9 represents nine
0.035?
thousandths, which is written as 0.009
 the number 5.239 in words is ‘five point two
Why do 25  10 and 250  100 give the same
three nine’ not ‘five point two hundred and
answer?
thirty nine’
My calculator display shows 0.001. Tell me
 the fraction 5 239 is read as ‘five and two
1000
what will happen when I multiply by 100. What
hundred and thirty-nine thousandths.
will the display show?
Complete statements such as:
4  10 = 
4   = 0.04
0.4 x 10 = 
0.4 x  = 400
0.4  10 = 
0.4  = 0.004
 100 = 0.04
I divide a number by 10, and then again by 10.
The answer is 0.3. What number did I start with?
How do you know?
How would you explain how to multiply a
decimal by 10 …., how to divide a decimal by
100?
Ratio and Proportion
Begin to understand simple ratio
Pupils use the vocabulary of ratio to describe
the relationships between two quantities within a
context
Given a bag of 4 red and 20 blue cubes, write
the ratio of red cubes to blue cubes, and the
ratio of blue cubes to red cubes
Solve simple problems using informal strategies:
For example:
A girl spent her savings of £40 on books and
clothes in the ratio 1:3. How much did she
spend on clothes?
Scale numbers up or down, for example, by
converting recipes for, say, 6 people to recipes
for 2 people:
In a recipe for 6 people you need 120 g flour and
270 ml of milk. How much of each ingredient
does a recipe for 2 people require?
Level 4
Can you talk me through how you solved this
problem?
What do you see as the important information in
this problem? How do you use it to solve the
problem?
Understand simple ratio
Write 16 :12 in its simplest form
Solve problems such as:
28 pupils are going on a visit. They are in the
ratio of 3 girls to 4 boys. How many boys are
there??
Level 5
How do you know when a ratio is in its simplest
form?
Is the ratio 1:5 the same as the ratio 5:1?
Explain your answer.
Convince me that 19:95 is the same ratio as 1:5
The instructions on a packet of cement say, ‘mix
sand and cement in the ratio 5:1’. A builder
mixes 5 kg of cement with 1 bucketful of sand.
Could this be correct? Explain your answer.
The ratio of boys to girls at a school club is 1:2.
Could there be 10 pupils at the club altogether?
Explain your answer.
Solve simple problems involving ratio and direct proportion
The ratio of yogurt to fruit puree used in a recipe How do you decide how to link the numbers in
is 5 : 2. If you have 200g of fruit puree, how
the problem with a given ratio? How does this
much yogurt do you need? If you have 250g of
help you to solve the problem?
yogurt, how much fruit puree do you need?
The ratio of boys to girls in a class is 4 : 5. How
A number of cubes are arranged in a pattern and many pupils could be in the class? How do you
the ratio of red cubes to green cubes is 2 : 7. If
know?
the pattern is continued until there are 28 green
cubes, how many red cubes will there be?
Give pupils several different simple problems
and ask:
Three bars of chocolate cost 90p. How much
Which of these problems are linked to, for
would six bars cost? And twelve bars?
example the ratio 2: 3. How do you know?
Six stuffed peppers cost £9.
What will 9 stuffed peppers cost?
Divide a quantity into two or more parts in a given ratio
Level 6
and solve problems involving ratio and direct proportion
Solve problem such as:
If the ratio of boys to girls in a class is 3:1, could
 Potting compost is made from loam, peat
there be exactly 30 children in the class? Why?
and sand in the ratio 7:3:2 respectively. A
Could there be 25 boys? Why?
gardener used 1.5 litres of peat to make
compost. How much loam did she use?
5 miles is about the same as 8km.
How much sand? ·
 Can you make up some conversion
 The angles in a triangle are in the ratio 6:5:7.
questions that you could answer mentally?
Find the sizes of the three angles.
 Can you make up some conversion
questions for which you would have to use a
more formal method?
How would you work out the answers to these
questions?
Use proportional reasoning to solve a problem, choosing the correct numbers to take as
100%, or as a whole
Use unitary methods and multiplicative methods, Which are the key words in this problem? How
e.g.
do these words help you to decide what to do?
 There was a 25% discount in a sale. A boy
paid £30 for a pair of jeans in the sale. What What are the important numbers? What are the
was the original price of the jeans?
important links that might help you solve the
 When heated, a metal bar increases in
problem?
length from 1.255m to 1.262m. Calculate the
percentage increase correct to one decimal
How do you decide which number represents
place.
100% or a whole when working on problems?
A recipe for fruit squash for six people is:
300g chopped oranges
1500ml lemonade
750ml orange juice
Trina made fruit squash for ten people. How
many millilitres of lemonade did she use?
Jim used two litres of orange juice for the same
recipe. How many people was this enough for?
Do you expect the answer to be larger or
smaller? Why?
What would you estimate the answer to be?
Why?
Understand and use proportionality
Examples of what pupils know and be able to do
Sets A, B and C are in direct proportion.
Set A
5
3
7
Set B
Set C
11
5
2.4
1.1
17.6
10.5
 Is there sufficient information to find all the
missing entries?
 What is the maximum number of items which
could be entered and the task remain
impossible?
 What is the minimum number of entries
needed and what is important about their
location?
 For any one empty cell – What is the best
starting point? How many different starting
points are there?
Level 7
Probing questions
How do you go about finding the missing
numbers in this table?
Miles
5
?
36
Kilometres
8
20
?
Can you do it in a different way? What do you
look for in deciding the most efficient way to find
the missing numbers?
How do you go about checking whether given
sets of information are in direct proportion?
What hints would you give someone to help
them solve word problems involving
proportionality?
Talk me through the reasoning that took you to
this answer.
How did you think this through?
Is a 50% increase followed by a 50% increase
the same as doubling? Explain your answer.
Which is better? A 70% discount or a 50%
discount and a further 20% special offer
discount?
A shop is offering a 10% discount. Is it better to
have this before or after VAT is added at 17.5%?
Which is the better deal? Buy one get one half
price or three for the price of two?
Calculate the result of any proportional change using multiplicative methods
Examples of what pupils know and be able to do Probing questions
The new model of an MP3 player holds1/6 more How do you go about finding a multiplier to
music than the previous model. The previous
increase by a given fraction (percentage)? What
model holds 5000 tracks. How many tracks
if it was a fractional (percentage) decrease?
does the new model hold?
The previous model cost £119.99 and the new
Why is it important to identify ‘the whole’ when
model costs £144.99. Is this less than or greater working with problems involving proportional
than the proportional change to the number of
change?
tracks? Justify your answer.
How do you go about finding a multiplier to
After one year a scooter has depreciated by 1/7
calculate an original value after a proportional
and is valued at £995. What was its value at the increase/decrease?
beginning of the year?
Given a multiplier how can you tell whether this
Weekend restaurant waiting staff get a 4%
would result in an increase or a decrease?
increase. The new hourly rate is £4.94. What
was it before the increase?
My friend’s savings amount to £920 after 7%
interest has been added. What was the original
amount of her savings before interest was
added?
Use a multiplicative methods such as:
Original
Result
Multiplier
100%
107%
x 100/107
?
£920
x 100/107
Types of Number
Recognise and describe number relationships including multiple,
factor and square
(Level 4)
Use the multiples of 4 to work out the multiples
Which numbers less than 100 have exactly three
of 8.
factors?
Identify factors of two-digit numbers
What number up to 100 has the most factors?
Know simple tests for divisibility for 2, 3, 4, 5, 6,
8, 9
The sum of four even numbers is a multiple of
four. When is this statement true? When is it
false?
Find the factors of a number by checking for
divisibility by primes. For example, to find the
factors of 123 check mentally or otherwise, if the
number divides by 2, then 3,5,7,11…
Can a prime number be a multiple of 4? Why?
Can you give me an example of a number
greater than 500 that is divisible by 3? How do
you know?
How do you know if a number is divisible by 6?
etc
Can you give me an example of a number
greater that 100 that is divisible by 5 and also by
3? How do you know?
Is there a quick way to check if a number is
divisible by 25?
Recognise and use number patterns and relationships
(Level 5)
Find:
 A prime number greater than 100
 The largest cube smaller than 1000
 Two prime numbers that add to 98
Give reasons why none of the following are
prime numbers:
4094, 1235, 5121
Use factors, when appropriate, to calculate
mentally, e.g.
35 × 12 = 35 × 2 × 6
Continue these sequences:
 8, 15, 22, 29, …
 6, 2, -2, -6, …
 1, 1⁄2, 1⁄4, 1⁄8,
 1, -2, 4, -8
 1, 0.5, 0.25
 1, 1, 2, 3, 5, 8
Talk me through an easy way to do this
multiplication/division mentally. Why is
knowledge of factors important for this?
How do you go about identifying the factors of a
number greater than 100?
What is the same / different about these
sequences:
 4.3, 4.6, 4.9, 5.2, …
 16.8, 17.1, 17.4, 17.7, …
 9.4, 9.1, 8.8, 8.5, ...
I’ve got a number sequence in my head. How
many questions would you need to ask me to be
sure you know my number sequence? What are
the questions?
3D Shape
Make 3-D models by linking given faces or edges and
draw common 2-D shapes in different orientations on grids
Identify and make up the different nets for an When presented with a net:
open cube.
 Which edge will meet this edge?
 Which vertices will meet this one?
Level 4
What can you tell me about a 3-D shape from its 2-D
net?
Complete a rectangle which has 2 sides
drawn at an oblique angle to the grid
How would you plot a rectangle that has no
horizontal sides on a square grid? How would you
convince me that the shape is a rectangle?
Visualise and use 2-D representations of 3-D objects
Level 6
How would the 3-D shape be different if the plan was
Visualise solids from an oral description, e.g. a rectangle rather than a square? Why? Are there
Identify the 3-D shape if:
other possible 3-D shapes?
 The front and side elevations are both
triangles and the plan is a square.
Starting from a 2-D representation of a 3-D shape:
 The front and side elevations are both
 How many faces will the 3-D shape have? How
rectangles and the plan is a circle.
do you know?
 The front elevation is a rectangle, the
 What will be opposite this face in the 3-D shape?
side elevation is a triangle and the plan
How do you know?
in a rectangle.
 Which side will this side join to make an edge?
How do you know?
Is it possible to slice a cube so that the
 How would you go about drawing the plan and
cross-section is:
elevation for the 3-D shape you could make from
this net?

a rectangle?

a triangle?

a pentagon?

a hexagon?
Calculate lengths, areas and volumes in plane shapes and
Level 7
right prisms
The cross section of a skirting board is the
Talk me through the steps you took when finding the
shape of a rectangle, with a quadrant
surface area of this right prism.
(quarter circle) on the top. The skirting
board is 1.5cm thick and 6.5cm high.
If you know the height and volume of a right prism,
Lengths totalling 120m are ordered. What
what else do you know? What don’t you know?
volume of wood is contained in the order?
How many different square-based right prisms have
a height of 10 cm and a volume of 160 cm3? Why?
What do you need to know to be able the find both
the volume and surface area of a cylinder?
Angles and Triangles
Use the properties of 2-D and 3-D shapes
Recognise and name most quadrilaterals
e.g. trapezium, parallelogram, rhombus
Recognise right-angled, equilateral,
isosceles and scalene triangles and know
their properties
Use mathematical terms such as
horizontal, vertical, parallel, perpendicular
Level 4
Can you describe a rectangle precisely in
words so someone else can draw it?
What mathematical words are important
when describing a rectangle?
What properties do you need to be sure a
triangle is isosceles; equilateral; scalene?
Show a mix of rectangles including squares.
Can you tell me which of the shapes are
square? Convince me?
Can you tell me which shapes are
rectangles? Convince me?
Understand properties of shapes, e.g. why
a square is a special rectangle
Visualise shapes and recognise them in
different orientations
Use language associated with angle and know and use the
Level 5
angle sum of a triangle and that of angles at a point
Calculate ‘missing angles’ in triangles
Is it possible to draw a triangle with:
including isosceles triangles or right angled i) one acute angle
triangles, when only one other angle is
ii) two acute angles
given
iii) one obtuse angle
iv) two obtuse angles
Calculate angles on a straight line or at a
Give an example of the three angles if it is
point, such as the angle between the
possible. Explain why if it is impossible.
hands of a clock, or intersecting diagonals
at the centre of a regular hexagon
Explain why a triangle cannot have two
parallel sides.
Understand ‘parallel’ and ‘perpendicular’ in
relation to edges or faces of 2-D shapes
How can you use the fact that the sum of the
angles on a straight line is 180º to explain
why the angles at a point are 360º?
An isosceles triangle has one angle of 30º. Is
this enough information to know the other two
angles? Why?
Identify alternate and corresponding angles; understand a proof
Level 6
that the sum of the angles of a triangle is 180° and
of a quadrilateral is 360°
Know the difference between a
How could you convince me that the sum of
demonstration and a proof.
the angles of a triangle is 180º?
Understand a proof that the sum of the
angles of a triangle is 1800 and of a
quadrilateral is 3600.
Why are parallel lines important when proving
the sum of the angles of a triangle?
How does knowing the sum of the interior
angles of a triangle help you to find the sum
of the interior angles of a quadrilateral? Will
this work for all quadrilaterals? Why?
Area and Perimeter
Find perimeters of simple shapes and find areas by counting squares
Level 4
Find perimeters and areas of shapes
How do you go about finding the perimeter of a
other than rectangles. Focus on having a shape?
feel for the perimeter and area - not
calculating them.
How are the perimeter of a shape and the area
of a shape different? How do you remember
Work out the perimeter of some shapes
which is which?
by measuring, in millimetres.
Would you expect the area of a paperback
Use the terms area and perimeter
book cover to be: 200cm², 600cm², or
accurately and consistently
6000cm²? Explain why.
Find areas by counting squares and part
squares of shapes drawn on squared
paper.
Begin to find the area of shapes that are
made from joining rectangles
Would you expect the area of a digit card to be:
5 cm², 50cm² or 100cm²? Explain why.
Suggest 2-D shapes/objects where the area
could be measured in cm².
Use ‘number of squares in a row times
number of rows’ to find the area of a
rectangle
Understand and use the formula for the area of a rectangle and
Level 5 distinguish area from perimeter
Find any one of the area, width and
For a given area (e.g. 24cm2) find as many
length of a rectangle, given the other
possible rectangles with whole number
two.
dimensions as you can. How did you do it?
Find any one of the perimeter, width and
length of a rectangle, given the other
two.
Find the area or perimeter of simple
compound shapes made from rectangles
The carpet in Walt’s living room is
square, and has an area of 4 m2. The
carpet in his hall has the same perimeter
as the living room carpet, but only 75%
of the area. What are the dimensions of
the hall carpet?
For compound shapes formed from rectangles:
How do you go about finding the dimensions
needed to calculate the area of this shape?
Are there other ways to do it? How do you go
about finding the perimeter?
Always, sometimes or never true?
 If one rectangle has a larger perimeter than
another one, then it will also have a larger
area.
Deduce and use formulae for the area of a triangle and
Level 6
parallelogram, and the volume of a cuboid; calculate
volumes and surface areas of cuboids
Calculate the areas of triangles and
Why do you have to multiply the base by the
parallelograms
perpendicular height to find the area of a
parallelogram?
Suggest possible dimensions for
triangles and parallelograms when the
The area of a triangle is 12cm2. What are the
area is known.
possible lengths of base and height?
Calculate the volume and the surface
area of a 3cm by 4 cm by 5cm box.
Find three cuboids with a volume of
24cm3
Find a cuboid with a surface area of
60cm2
Right-angled triangles have half the area of the
rectangle with the same base and height. What
about non-right-angled triangles?
What other formulae for the area of 2-D shapes
do you know? Is there a formula for the area of
every 2-D shape?
How do you go about finding the volume of a
cuboid? How do you go about finding the
surface area of a cuboid?
‘You can build a solid cuboid using any number
of interlocking cubes.’ Is this statement always,
sometimes or never true? If it is sometimes
true, when is it true and when is it false? For
what numbers can you only make one cuboid?
For what numbers can you make several
different cuboids?
Know and use the formulae for the circumference and area of a circle
A circle has a circumference of 120cm.
What is the minimum information you need to
What is the radius of the circle?
be able to find the circumference and area of a
circle?
A touring cycle has wheels of diameter
70cm. How many rotations does each
Give pupils some work with mistakes. Ask
wheel make for every 10km travelled?
them to identify and correct the mistakes.
A circle has a radius of 15cm. What is its
area?
A door is in the shape of a rectangle with
a semi-circular arch on top. The
rectangular part is 2m high and the door
is 90cm wide. What is the area of the
door?
How would you go about finding the area of a
circle if you know the circumference?
Calculate lengths, areas and volumes in plane shapes and
right prisms
The cross section of a skirting board is
the shape of a rectangle, with a quadrant
(quarter circle) on the top. The skirting
board is 1.5cm thick and 6.5cm high.
Lengths totalling 120m are ordered.
What volume of wood is contained in the
order?
Level 7
Talk me through the steps you took when
finding the surface area of this right prism.
If you know the height and volume of a right
prism, what else do you know? What don’t you
know?
How many different square-based right prisms
have a height of 10 cm and a volume of 160
cm3? Why?
What do you need to know to be able the find
both the volume and surface area of a cylinder?
Understand the difference between
Level 8
formula for perimeter, area and volume in simple contexts by
considering dimensions
Work with formulae for a range of 2-D
How do you go about deciding whether a
and 3-D shapes and relate the results
formula is for a perimeter, an area or a volume?
and dimensions.
Why is it easy to distinguish between the
formulae for the circumference and area of a
circle?
How would you help someone to distinguish
between the formula for the surface area of a
cube and the volume of a cube?
How do you decide whether a number in a
calculation represents the length of a
dimension?
Compound Measures
Understand and use measures of speed
Level 7
(and other compound measures such as
density or pressure) to solve problems
Use examples of compound measures in
science, geography and PE.
Understand that:
 Rate is a way of comparing how one
quantity changes with another, e.g. a
car’s fuel consumption measured in miles
per gallon.
 The two quantities are usually measured
in different units, and ‘per’, the
abbreviation ‘p’ or an oblique ‘/’ can be
used to mean ‘for every’ or ‘in every’.
Solve problems such as:
The distance from London to Leeds is 190
miles. An intercity train takes about 2 ¼
hours to travel from London to Leeds. What
is its average speed?
.
Make up some easy questions that involve
calculating speed, distance or time (density,
mass and volume). Make up some difficult
questions. What makes them difficult?
Talk me through the reasoning of why
travelling a distance of 30 miles in 45
minutes is an average of 40mph.
How do the units of speed (density,
pressure) help you to solve problems?
How does the information in a question help
you to decide on the units for speed (density,
pressure)?
Congruence and Similarity
Understand and use congruence and mathematical similarity
Level 8
Understand and use the preservation of the
What do you look for when deciding whether
ratio of side lengths in problems involving
two triangles are congruent?
similar shapes.
What do you look for when deciding whether
Use congruent triangles to prove that the two two triangles are similar?
base angles of an isosceles triangle are
equal by drawing the perpendicular bisector
Which of these statements are true? Explain
of the base.
your reasoning.
 Any two right angled triangles will be
Use congruence to prove that the diagonals
similar
of a rhombus bisect each other at right
 If you enlarge a shape you get two
angles.
similar shapes
 All circles are similar
Convince me that:
 Any two regular polygons with the same
number of sides are similar
 Alternate angles are equal (using
congruent triangles)
Dimensions
Understand the difference between formulae for perimeter,
Level 8
area and volume in simple contexts by considering dimensions
Work with formulae for a range of 2-D and 3- How do you go about deciding whether a
D shapes and relate the results and
formula is for a perimeter, an area or a
dimensions.
volume?
Why is it easy to distinguish between the
formulae for the circumference and area of a
circle?
How would you help someone to distinguish
between the formula for the surface area of a
cube and the volume of a cube?
How do you decide whether a number in a
calculation represents the length of a
dimension?
Locus and Constructions
Use straight edge and compasses to do standard constructions
Level 6
Use straight edge and compasses to construct:
Why are compasses important when doing
 the mid-point and perpendicular bisector of a constructions?
line segment
 the bisector of an angle
How do the properties of a rhombus help with
 the perpendicular from a point to a line
simple constructions such as bisecting an angle?
segment
 the perpendicular from a point on a line
For which constructions is it important to keep the
segment.
same compass arc (distance between the pencil
and the point of your compasses)? Why?
Construct triangles to scale using ruler and
protractor (SAS, ASA)and using straight edge
and compasses (SSS)
Find the locus of a point that moves according to a given rule,
Level 7
both by reasoning and using ICT.
Visualise the result of spinning 2D shapes in 3D
How can you tell for a given locus whether it is the
around a line that is along a line of symmetry of
path of points equidistant from another point or a
the shape.
line?
Trace the path of a vertex of a square as it is
rolled along a straight line.
Visualise simple paths such as that generated by
walking so that you are equidistant from two
trees.
Find the locus of:
 points equidistant from two points,
 points equidistant from a line,
 points equidistant from a point
 the centre of circles which have two given
lines as tangents
What is the same/different about the path traced
out by the centre of a circle being rolled along a
straight line and the centre of a square being
rolled along a straight line?
Properties of Shapes
Use the properties of 2-D and 3-D shapes
Recognise and name most quadrilaterals
e.g. trapezium, parallelogram, rhombus
Recognise right-angled, equilateral,
isosceles and scalene triangles and know
their properties
Use mathematical terms such as
horizontal, vertical, parallel, perpendicular
Level 4
Can you describe a rectangle precisely in words so
someone else can draw it?
What mathematical words are important when
describing a rectangle?
What properties do you need to be sure a triangle is
isosceles; equilateral; scalene?
Show a mix of rectangles including squares.
Can you tell me which of the shapes are square?
Convince me?
Can you tell me which shapes are rectangles?
Convince me?
Understand properties of shapes, e.g.
why a square is a special rectangle
Visualise shapes and recognise them in
different orientations
Use a wider range of properties of 2-D and 3-D shapes
Level 5
and identify all the symmetries of 2-D shapes
Understand ‘parallel’ and ‘perpendicular’
Sketch me a quadrilateral that has one line of
in relation to edges and faces of 3-D
symmetry, two lines, three lines, no lines etc. Can you
shapes.
give me any others? What is the order of rotational
symmetry of each of the quadrilaterals you sketched?
Find lines of symmetry in 2-D shapes
including oblique lines.
One of the lines of symmetry of a regular polygon
goes through two vertices of the polygon. Convince
Recognise the rotational symmetry of
me that the polygon must have an even number of
familiar shapes, such as parallelograms
sides.
and regular polygons.
Sketch a shape to help convince me that:
 A trapezium might not be a parallelogram
Classify quadrilaterals, including
 A trapezium might not have a line of symmetry
trapeziums, using properties such as
 Every parallelogram is also a trapezium
number of pairs of parallel sides.
Classify quadrilaterals by their geometric properties
Know the properties (equal and/or parallel What properties do you need to know about a
sides, equal angles, right angles,
quadrilateral to be sure it is a kite; a parallelogram; a
diagonals bisected and/or at right angles,
rhombus; an isosceles trapezium?
reflection and rotation symmetry) of:
 an isosceles trapezium
Can you convince me that a rhombus is a
 a parallelogram
parallelogram but a parallelogram is not necessarily a
 a rhombus
rhombus?
 a kite
 an arrowhead or delta
Why can’t a trapezium have three acute angles?
Which quadrilateral can have three acute angles?
Pythagoras
Level 7
Understand and apply Pythagoras' theorem when solving problems in 2-D
Know that Pythagoras’ Theorem only holds for
triangles that are right-angled.
How do you identify the hypotenuse when
solving a problem using Pythagoras’ theorem?
Identify a right angled triangle in a problem
What do you look for in a problem to decide
whether it can be solved using Pythagoras’
theorem?
Find a missing side in a right-angled triangle.
For example:
Use Pythagoras’ theorem to solve simple
problems in two dimensions, such as
 A 5m ladder leans against a wall with its foot
1.5m away from the wall. How far up the wall
does the ladder reach
 You walk due north for 5 miles, then due east
for 3 miles. What is the shortest distance you
are from your starting point?
Identify triangles that must be right-angled from
their side-lengths
Talk me through how you went about drawing
and labelling this triangle for this problem.
How can you use Pythagoras’ theorem to tell
whether an angle in a triangle is equal to, greater
than or less than 90 degrees?
What is the same/different about a triangle with
sides 5cm, 12cm and an unknown hypotenuse
and a triangle with sides 5cm, 12cm and an
unknown shorter side?
Level 8 (from trig section)
Sketch right angled triangles for problems
expressed in words.
How do you decide whether a problem requires
use of a trigonometric relationship (sine, cosine
or tangent) or Pythagoras’ theorem to solve it?
Transformations
Reflect simple shapes in a mirror line, translate shapes horizontally or vertically
and begin to rotate a simple shape or object about its centre or a vertex
Level 4
Use a grid to plot the reflection in a mirror line
Give me instructions to reflect this shape into
presented at 45° for both where the shape
this mirror line. What would you suggest I do
touches the mirror line and where it does not
first?
Begin to use the distance of vertices from the
mirror line to reflect shapes more accurately
How do the squares on the grid help when
reflecting? Show me.
Make up a reflection that is easy to do.
Make up a reflection that is hard to do. What
makes it hard?
How can you tell if a shape has been reflected or
translated?
Reason about position and movement and transform shapes
Level 5
Construct the reflections of shapes in mirror
Make up a reflection/rotation that is easy to do.
lines placed at different angles relative to the
shape:
Make up a reflection/rotation that is hard to do.
What makes it hard?
 Reflect shapes in oblique (45°) mirror lines
where the shape either does not touch the
What clues do you look for when deciding
mirror line, or where the shape crosses the
whether a shape has been reflected or rotated?
mirror line
 Reflect shapes not presented on grids, by
measuring perpendicular distances to/from
What transformations can you find in patterns in
the mirror.
flooring, tiling, wallpaper, wrapping paper…?
Reflect shapes in two mirror lines, where the
What information is important when describing a
shape is not parallel or perpendicular to either
reflection/rotation?
mirror.
Rotate shapes, through 90° or 180°, when the
centre of rotation is a vertex of the shape, and
recognise such rotations.
Describe how rotating a rectangle about its
centre looks different from rotating it about one
of its vertices.
Translate shapes along an oblique line.
How would you describe this translation
precisely?
Reason about shapes, positions and
movements, e.g.
 visualise a 3-D shape from its net and match
vertices that will be joined
 visualise where patterns drawn on a 3-D
shape will occur on its net
Enlarge 2-D shapes, given a centre of enlargement
and a positive whole-number scale factor
Level 5
Construct an enlargement given the object,
What changes when you enlarge a shape?
centre of enlargement and scale factor.
What stays the same?
Find the centre of enlargement and/or scale
factor from the object and image.
What information do you need to complete a
given enlargement?
If someone has completed an enlargement how
would you find the centre and the scale factor?
When doing an enlargement, what strategies do
you use to make sure your enlarged shape will
fit on the paper?
Know that translations, rotations and reflections preserve length
and angle and map objects onto congruent images
Level 6
Find missing lengths and angles on diagrams
What changes and what stays the same when
that show an object and its image
you:
 translate
Match corresponding lengths and angles of
 rotate
object and image shapes following reflection,
 reflect
translation and/or rotation or a combination of
a shape?
these.
When is the image congruent? How do you
know?
Enlarge 2-D shapes, given a centre of enlargement and a fractional scale
factor,
on paper and using ICT; recognise the similarity of the resulting shapes
Level 7
Given an object and its enlargement what can
Enlarge a simple shape on suared paper by a
you say about the scale factor? How would you
fractional scale factor, such as ½ or 1/3 and
recognise that the scale factor is a fraction
recognise that the ratio of any two corresponding between 0 and 1?
sides is equal to the scale factor.
How would you go about finding the centre of
Use dynamic geometry software to explore
enlargement and the scale factor for two similar
enlargements, e.g. changing the scale factor or
shapes?
the centre of enlargement.
How does the position of the centre of
Investigate the standard paper sizes A1, A2, A3, enlargement (e.g. inside, on a vertex, on a side,
exploring the ratio of the sides of any A sized
or outside the original shape) affect the image?
paper and the scale factors between different A
How is this different if the scale factor is between
sized papers?
0 and 1?
Trigonometry
Understand and use trigonometrical relationships
Level 8
in right-angled triangles, and use these to solve
problems, including those involving bearings
Use sine, cosine and tangent as ratios (link to
similarity)
Is it possible to have a triangle with the
angles and lengths shown below?
Find missing sides in problems involving rightangled triangles in two dimensions
Find missing angles in problems involving rightangled triangles in two dimensions
Sketch right angled triangles for problems
expressed in words.
Solve problems such as,
Calculate the shortest distance between the
buoy and the harbour and the bearing that
the boat sails on. The boat sails in a
straight line from the harbour to the buoy.
The buoy is 6km to the east 4km to the
north of the harbour.
What do you look for when deciding
whether a problem can be solved using
trigonometry?
What’s the minimum information you
need about a triangle to be able to
calculate all three sides and all three
angles?
How do you decide whether a problem
requires use of a trigonometric
relationship (sine, cosine or tangent) or
Pythagoras’ theorem to solve it?
Is this statement always, sometimes or
never true? You can use trigonometry to
find missing lengths and/or angles in all
triangles.
Why is it important to understand similar
triangles when using trigonometric
relationships (sine, cosines and
tangents)?
Units and Reading Scales
Choose and use appropriate units and instruments
Level 4
Know metric conversions: mm/cm , cm/m , m/km, How do the names of units like millimetres,
mg/g , g/kg, ml/l
centimetres, metres, kilometres help you to
convert from one unit to another?
How do you go about finding the perimeter of a
rectangle when one side is measured in cm
and the other in mm?
When is it essential to use a ruler rather than a
straight edge?
Interpret, with appropriate accuracy, numbers on a range of measuring instruments
Measure and draw lengths and angles accurately What is the first thing you look for when you are
(±2mm ±5º)
reading a scale on measuring equipment?
Read and interpret scales on a range of
measuring instruments, including:
– vertical scales, e.g. thermometer, tape
measure, ruler…
– scales around a circle or semi-circle, e.g. for
measuring time, mass, angle…
How do you decide what each division on the
scale represents?
How would you measure 3.6 cm if the zero end
of your ruler was broken?
Read and interpret scales on a range of measuring instruments,
Level 5
explaining what each labelled division represents
Read and interpret scales on a variety of real
When reading scales how do you decide what
measuring instruments and illustrations; for
each division on the scale represents?
example, rulers and tape measures, spring
balances and weighing scales, thermometers,
What mistakes could somebody make when
car instruments and electrical meters.
reading from a scale? How would you avoid
these mistakes?
Explain what each labelled division represents on
a scale.
Solve problems involving the conversion of units and make sensible estimates of a range of
measures in relation to everyday situations
Change a larger unit into a smaller one. e.g.
Which is longer 200cm or 20 000mm? Explain
 Change 36 centilitres into millilitres
how you worked it out.
 Change 0.89km into metres
 Change 0.56 litres into millilitres
Give me another length that is the same as
3m.
Change a smaller unit into a larger one. e.g.
 Change 750 g into kilograms
What clues do you look for when deciding
 Change 237 ml into litres
which metric unit is bigger?
 Change 3 cm into metres
 Change 4mm into centimetres
Explain how you convert metres to
centimetres.
Solve problems such as:
How many 30g blocks of chocolate will weigh
How do you change g into kilograms, ml into
1.5kg; using 1.5kg ÷ 30g ?
litres, km into metres etc?
Know rough metric equivalents of imperial
measures in daily use (feet, miles, pounds, pints,
gallons).
Work out approximately how many km are
equivalent to 20 miles.
What rough metric equivalents of imperial
measurements do you know?
How would you change metres into feet, km
into miles etc? What do you need to know to
be able to do this?